Finiteness and upper bound computations for the four-vortex problem

Introduction

This notebook contains computational algorithms and results which support the paper "Finiteness of Stationary Configurations of the Four-Vortex Problem" by Marshall Hampton and Richard Moeckel. The notebook is divided into sections explaining the different kinds of computations used to verify the results in the paper. The reader can go through the notebook section by section, executing the cells in order to check the computations.

Equilibria

Section 2 of the paper is about the equilibria of the four-vortex problem.  Here are some commands to set up the equations.

VortexIVBar[i_, n_]  :=  
      Plus @@ (Table[G[j]/(z[i] - z[j]), {j, 1, i - 1}]~Join~
            Table[G[j]/(z[i] - z[j]), {j, i + 1, 
                n}])


Denoms[i_, n_]  :=  
      Times @@ (Table[(z[i] - z[j]), {j, i + 1, n}]~Join~
            Table[(z[i] - z[j]), {j, 1, 
                i - 1}])


VortexEquilibriaEqs[
        n_]  :=  Table[Denoms[i, n]*VortexIVBar[i, n], {i, 1, n}] // 
          Factor // Numerator

VortexEquilibriaEqs0[
        n_]  :=  Table[
              Denoms[i, n]*VortexIVBar[i, n], {i, 3, n}] /. {z[n - 1] -> 
                1, z[n] -> 0} // Factor // 
        Numerator

Here are the two normalized equations using the notation of the paper.

{F3, F4} =  VortexEquilibriaEqs0[4]

The resultant with respect to z2.  The coefficients are simplified using the equation L=0

rz1  =   Resultant[F3, F4, z[2]] // 
        Factor

clrz1  =  
      CoefficientList[rz1, z[1]] // Factor
L = 
        G[1] G[2] + G[1] G[3] + G[1] G[4] + G[2] G[3] + G[2] G[4] + 
          G[3] G[4];
clrz1[[2]]  =  
        clrz1[[2]] + L // Factor;
clrz1[[3]]  =  
        clrz1[[3]] - L // 
          Factor;

rz1 =  clrz1 . {1, z[1], z[1]^2}

{a, b, c}  =  
      clrz1/(-G[1]) // 
        Factor

disc  = b^2 - 4*a*c // Expand
disc  =  
      disc + 4  L // Factor

z1sol  =  (-b + I*Sqrt[3]*G[2])/(2*a)

Rigidly Translating Solutions

These calculations reproduce the computer-assisted parts of section 3, which shows that there at most 6 rigidly translating solutions of the four-vortex problem.

\(VarNames[n_] := 
      Flatten[Table[{z[j] -> 
              ToExpression[StringJoin["z", ToString[j]]], 
            G[j] -> ToExpression[StringJoin["G", ToString[j]]]}, {j, 
            1, n}]]; 
    G[i_, n_] := \((Sum[
              If[j ≠ i, 
                G[j] Product[
                    If[k ≠ j && k ≠ i, z[i] - z[k], 
                      1], {k, 1, n}] 
                  Product[
                    If[k ≠ i && k ≠ n, z[n] - z[k], 
                      1], {k, 1, n}], 0], {j, 1, n}]
	 + \ 
            Sum[If[j ≠ n, 
                G[j] Product[
                    If[k ≠ i && k ≠ n, z[i] - z[k], 
                      1], {k, 1, n}] 
                  Product[
                    If[k ≠ j && k ≠ n, z[n] - z[k], 
                      1], {k, 1, n}], 0], {j, 1, n}])\) /. 
        VarNames[n]\)

Here are the euqations called G2 and G3 in the paper.

RigidTranslateSystem = 
      Factor[{G[2, 4], G[3, 4]} /. {z4 -> 0, z3 -> 1}]

After dividing out a trivial factor, the resultant with respect to z2 gives a degree 6 polynomial in z1.

Factor[
      Resultant[RigidTranslateSystem[[1]], RigidTranslateSystem[[2]], 
        z2]]
p6 = 
        Factor[Resultant[RigidTranslateSystem[[1]], 
              RigidTranslateSystem[[2]], 
              z2]]/((-1 + z1) (z1^3) );

The next computation shows that the polynomial p6 cannot vanish identically for any relevant values of the vorticities.

RigidTranslateCoefficientIdeal = 
      Flatten[{G1 + G2 + G3 + G4, G1*G2*G3*G4 - 1, 
          Table[Coefficient[p6, z1, i], {i, 0, 
              6}]}]

GroebnerBasis[
      RigidTranslateCoefficientIdeal, {G1, G2, G3, G4}]

Collinear Relative Equilibria

The following calculations illustrate the computations in section 4, which show that there are at most 12 collinear relative equilibria in the four-vortex problem.  Here are the equations.

H[i_, n_] := 
          G[i, n] + λ (z[i] - z[n]) 
                Product[
                  If[k ≠ i && k ≠ n, z[i] - z[k], 1], {k, 
                    1, n}] Product[
                  If[k ≠ n, z[n] - z[k], 1], {k, 1, n}] /. 
            VarNames[n]

CollinearEqs = {H[1, 4], H[2, 4], H[3, 4]} /. {z4 -> 0, 
            z3 -> 1}

Next we eliminate l and divide out some trivial factors.

CollinearEqsResultants = 
      Flatten[Table[
          Factor[Resultant[CollinearEqs[[i]], 
              CollinearEqs[[j]], λ]], {i, 1, 2}, {j, i + 1, 
            3}]]

{K2, K3} = 
      Numerator[
        Factor[{CollinearEqsResultants[[2]], 
              CollinearEqsResultants[[3]]}/((-1 + 
                    z2) (-1 + z1) z1 z2 )]]

A resultant leads to a twelfth degree polynomial for z1.

P12 = 
      Factor[Resultant[K2, K3, 
            z2]]/((((-1 + z1))^5) (z1^5) )

Verification that P12 does not vanish identically.

P12CoefficientTable = 
        Table[Factor[Coefficient[P12, z1, i]], {i, 0, 
            12}];

Factor[
      GroebnerBasis[
        Union[{G1 G2 G3 G4  - 1}, P12CoefficientTable], {G1, G2, G3, G4}, 
        MonomialOrder -> DegreeReverseLexicographic]]

Strictly Planar Relative Equilibria with nonzero total circulation

This section contains our calculations used in section 5.

Evaluate the small data cell below to define vertex and inequality lists for the final Minkowski sum polytope:

v1t9bH = {{0, 6, 12, 18, 14, 10}, {0, 6, 12, 18, 14, 14}, {0, 6, 
            12, 18, 16, 10}, {0, 6, 12, 18, 16, 14}, {0, 6, 14, 18, 14, 
            8}, {0, 6, 14, 18, 16, 8}, {0, 6, 14, 18, 16, 12}, {0, 6, 16, 18, 
            10, 10}, {0, 6, 16, 18, 10, 14}, {0, 6, 16, 18, 12, 14}, {0, 6, 
            18, 18, 10, 8}, {0, 6, 18, 18, 10, 12}, {0, 6, 18, 18, 12, 
            8}, {0, 6, 18, 18, 12, 12}, {0, 8, 10, 16, 18, 10}, {0, 8, 10, 
            16, 18, 14}, {0, 8, 10, 18, 14, 10}, {0, 8, 10, 18, 14, 14}, {0, 
            8, 12, 18, 16, 14}, {0, 8, 12, 20, 12, 8}, {0, 8, 12, 20, 12, 
            12}, {0, 8, 12, 20, 14, 8}, {0, 8, 12, 20, 14, 12}, {0, 8, 16, 
            18, 8, 10}, {0, 8, 16, 18, 8, 14}, {0, 8, 16, 18, 12, 14}, {0, 8, 
            16, 20, 8, 8}, {0, 8, 16, 20, 8, 12}, {0, 8, 16, 20, 10, 8}, {0, 
            8, 16, 20, 10, 12}, {0, 8, 18, 16, 8, 10}, {0, 8, 18, 16, 8, 
            14}, {0, 8, 18, 16, 10, 14}, {0, 8, 20, 16, 8, 8}, {0, 8, 20, 16, 
            8, 12}, {0, 8, 20, 16, 10, 8}, {0, 8, 20, 16, 10, 12}, {0, 10, 8, 
            14, 18, 10}, {0, 10, 8, 14, 18, 14}, {0, 10, 8, 18, 16, 10}, {0, 
            10, 8, 18, 16, 14}, {0, 10, 10, 14, 20, 8}, {0, 10, 10, 14, 20, 
            12}, {0, 10, 10, 16, 18, 14}, {0, 10, 10, 18, 16, 14}, {0, 10, 
            10, 20, 14, 8}, {0, 10, 10, 20, 14, 12}, {0, 10, 16, 18, 8, 
            14}, {0, 10, 16, 18, 10, 14}, {0, 10, 16, 20, 8, 8}, {0, 10, 16, 
            20, 8, 12}, {0, 10, 18, 16, 6, 10}, {0, 10, 18, 16, 6, 14}, {0, 
            10, 18, 16, 10, 14}, {0, 10, 18, 18, 6, 8}, {0, 10, 18, 18, 6, 
            12}, {0, 10, 20, 16, 8, 8}, {0, 10, 20, 16, 8, 12}, {0, 12, 6, 
            14, 18, 10}, {0, 12, 6, 14, 18, 14}, {0, 12, 6, 16, 18, 10}, {0, 
            12, 6, 16, 18, 14}, {0, 12, 8, 12, 20, 8}, {0, 12, 8, 12, 20, 
            12}, {0, 12, 8, 14, 20, 8}, {0, 12, 8, 14, 20, 12}, {0, 12, 8, 
            16, 18, 14}, {0, 12, 18, 16, 6, 14}, {0, 12, 18, 16, 8, 14}, {0, 
            12, 18, 18, 6, 8}, {0, 12, 18, 18, 6, 12}, {0, 12, 20, 12, 8, 
            8}, {0, 12, 20, 12, 8, 12}, {0, 14, 6, 14, 18, 8}, {0, 14, 6, 16, 
            18, 8}, {0, 14, 6, 16, 18, 12}, {0, 14, 18, 10, 8, 10}, {0, 14, 
            18, 10, 8, 14}, {0, 14, 18, 12, 6, 10}, {0, 14, 18, 12, 6, 
            14}, {0, 14, 18, 14, 6, 8}, {0, 14, 20, 10, 10, 8}, {0, 14, 20, 
            10, 10, 12}, {0, 14, 20, 12, 8, 8}, {0, 14, 20, 12, 8, 12}, {0, 
            16, 6, 10, 18, 10}, {0, 16, 6, 10, 18, 14}, {0, 16, 6, 12, 18, 
            14}, {0, 16, 8, 8, 18, 10}, {0, 16, 8, 8, 18, 14}, {0, 16, 8, 8, 
            20, 8}, {0, 16, 8, 8, 20, 12}, {0, 16, 8, 10, 20, 8}, {0, 16, 8, 
            10, 20, 12}, {0, 16, 8, 12, 18, 14}, {0, 16, 10, 8, 18, 14}, {0, 
            16, 10, 8, 20, 8}, {0, 16, 10, 8, 20, 12}, {0, 16, 10, 10, 18, 
            14}, {0, 16, 18, 8, 10, 10}, {0, 16, 18, 8, 10, 14}, {0, 16, 18, 
            10, 10, 14}, {0, 16, 18, 12, 6, 10}, {0, 16, 18, 12, 6, 14}, {0, 
            16, 18, 12, 8, 14}, {0, 16, 18, 14, 6, 8}, {0, 16, 18, 14, 6, 
            12}, {0, 18, 6, 10, 18, 8}, {0, 18, 6, 10, 18, 12}, {0, 18, 6, 
            12, 18, 8}, {0, 18, 6, 12, 18, 12}, {0, 18, 8, 8, 16, 10}, {0, 
            18, 8, 8, 16, 14}, {0, 18, 8, 10, 16, 14}, {0, 18, 10, 6, 16, 
            10}, {0, 18, 10, 6, 16, 14}, {0, 18, 10, 6, 18, 8}, {0, 18, 10, 
            6, 18, 12}, {0, 18, 10, 10, 16, 14}, {0, 18, 12, 6, 16, 14}, {0, 
            18, 12, 6, 18, 8}, {0, 18, 12, 6, 18, 12}, {0, 18, 12, 8, 16, 
            14}, {0, 18, 14, 6, 12, 10}, {0, 18, 14, 6, 12, 14}, {0, 18, 14, 
            6, 14, 8}, {0, 18, 14, 8, 10, 10}, {0, 18, 14, 8, 10, 14}, {0, 
            18, 16, 6, 12, 10}, {0, 18, 16, 6, 12, 14}, {0, 18, 16, 6, 14, 
            8}, {0, 18, 16, 6, 14, 12}, {0, 18, 16, 8, 12, 14}, {0, 18, 16, 
            10, 8, 10}, {0, 18, 16, 10, 8, 14}, {0, 18, 16, 10, 10, 14}, {0, 
            20, 8, 8, 16, 8}, {0, 20, 8, 8, 16, 12}, {0, 20, 8, 10, 16, 
            8}, {0, 20, 8, 10, 16, 12}, {0, 20, 10, 8, 16, 8}, {0, 20, 10, 8, 
            16, 12}, {0, 20, 12, 8, 12, 8}, {0, 20, 12, 8, 12, 12}, {0, 20, 
            14, 8, 12, 8}, {0, 20, 14, 8, 12, 12}, {0, 20, 14, 10, 10, 
            8}, {0, 20, 14, 10, 10, 12}, {6, 0, 12, 18, 10, 14}, {6, 0, 12, 
            18, 10, 16}, {6, 0, 12, 18, 14, 14}, {6, 0, 12, 18, 14, 16}, {6, 
            0, 14, 18, 8, 14}, {6, 0, 14, 18, 8, 16}, {6, 0, 14, 18, 12, 
            16}, {6, 0, 16, 18, 10, 10}, {6, 0, 16, 18, 14, 10}, {6, 0, 16, 
            18, 14, 12}, {6, 0, 18, 18, 8, 10}, {6, 0, 18, 18, 8, 12}, {6, 0, 
            18, 18, 12, 10}, {6, 0, 18, 18, 12, 12}, {6, 6, 12, 18, 14, 
            16}, {6, 6, 12, 18, 16, 14}, {6, 6, 12, 22, 10, 12}, {6, 6, 12, 
            22, 12, 10}, {6, 6, 16, 14, 8, 18}, {6, 6, 16, 14, 18, 8}, {6, 6, 
            16, 18, 12, 14}, {6, 6, 16, 18, 14, 12}, {6, 6, 16, 20, 10, 
            12}, {6, 6, 16, 20, 12, 10}, {6, 6, 20, 14, 8, 14}, {6, 6, 20, 
            14, 14, 8}, {6, 6, 20, 16, 10, 12}, {6, 6, 20, 16, 12, 10}, {6, 
            8, 8, 10, 22, 12}, {6, 8, 8, 12, 22, 8}, {6, 8, 8, 22, 10, 
            12}, {6, 8, 8, 22, 12, 8}, {6, 8, 10, 12, 22, 6}, {6, 8, 10, 12, 
            22, 10}, {6, 8, 10, 16, 18, 14}, {6, 8, 10, 18, 14, 16}, {6, 8, 
            10, 22, 10, 12}, {6, 8, 12, 10, 22, 6}, {6, 8, 12, 10, 22, 
            8}, {6, 8, 12, 18, 12, 16}, {6, 8, 12, 22, 8, 12}, {6, 8, 14, 16, 
            6, 18}, {6, 8, 14, 18, 8, 16}, {6, 8, 14, 18, 14, 14}, {6, 8, 14, 
            20, 6, 14}, {6, 8, 14, 22, 10, 10}, {6, 8, 18, 14, 0, 14}, {6, 8, 
            18, 14, 0, 16}, {6, 8, 18, 14, 8, 16}, {6, 8, 18, 16, 10, 
            14}, {6, 8, 18, 18, 0, 10}, {6, 8, 18, 18, 0, 12}, {6, 8, 22, 12, 
            8, 12}, {6, 8, 22, 12, 12, 8}, {6, 8, 22, 14, 10, 10}, {6, 10, 8, 
            10, 22, 12}, {6, 10, 8, 14, 18, 16}, {6, 10, 8, 18, 16, 14}, {6, 
            10, 8, 22, 12, 6}, {6, 10, 8, 22, 12, 10}, {6, 10, 12, 10, 22, 
            8}, {6, 10, 12, 18, 14, 14}, {6, 10, 12, 22, 10, 10}, {6, 10, 14, 
            18, 12, 14}, {6, 10, 14, 22, 8, 10}, {6, 10, 16, 18, 8, 14}, {6, 
            10, 16, 20, 6, 12}, {6, 10, 18, 12, 0, 14}, {6, 10, 18, 12, 0, 
            16}, {6, 10, 18, 16, 0, 10}, {6, 10, 20, 16, 6, 12}, {6, 10, 22, 
            8, 8, 12}, {6, 10, 22, 8, 12, 6}, {6, 10, 22, 8, 12, 8}, {6, 10, 
            22, 10, 8, 12}, {6, 10, 22, 10, 12, 8}, {6, 10, 22, 12, 6, 
            12}, {6, 10, 22, 12, 10, 10}, {6, 10, 22, 14, 8, 10}, {6, 12, 0, 
            10, 18, 14}, {6, 12, 0, 10, 18, 16}, {6, 12, 0, 14, 18, 14}, {6, 
            12, 0, 14, 18, 16}, {6, 12, 6, 10, 22, 12}, {6, 12, 6, 12, 22, 
            10}, {6, 12, 6, 14, 18, 16}, {6, 12, 6, 16, 18, 14}, {6, 12, 8, 
            8, 22, 12}, {6, 12, 8, 12, 18, 16}, {6, 12, 8, 22, 10, 6}, {6, 
            12, 8, 22, 10, 8}, {6, 12, 10, 10, 22, 10}, {6, 12, 10, 14, 18, 
            14}, {6, 12, 10, 22, 10, 8}, {6, 12, 12, 8, 22, 8}, {6, 12, 12, 
            22, 8, 8}, {6, 12, 16, 20, 6, 10}, {6, 12, 18, 12, 8, 16}, {6, 
            12, 18, 14, 0, 16}, {6, 12, 18, 14, 10, 14}, {6, 12, 18, 16, 6, 
            14}, {6, 12, 18, 18, 0, 10}, {6, 12, 18, 18, 0, 12}, {6, 12, 20, 
            16, 6, 10}, {6, 12, 22, 8, 8, 8}, {6, 12, 22, 8, 10, 6}, {6, 12, 
            22, 8, 10, 10}, {6, 12, 22, 12, 6, 10}, {6, 14, 0, 8, 18, 
            14}, {6, 14, 0, 8, 18, 16}, {6, 14, 0, 12, 18, 16}, {6, 14, 8, 6, 
            16, 18}, {6, 14, 8, 6, 20, 14}, {6, 14, 8, 8, 18, 16}, {6, 14, 8, 
            10, 22, 10}, {6, 14, 8, 14, 18, 14}, {6, 14, 10, 8, 22, 10}, {6, 
            14, 10, 12, 18, 14}, {6, 14, 14, 6, 20, 8}, {6, 14, 14, 20, 6, 
            8}, {6, 14, 18, 6, 16, 8}, {6, 14, 18, 10, 8, 16}, {6, 14, 18, 
            12, 0, 14}, {6, 14, 18, 12, 0, 16}, {6, 14, 18, 12, 6, 16}, {6, 
            14, 18, 12, 10, 14}, {6, 14, 18, 14, 8, 14}, {6, 14, 18, 16, 0, 
            10}, {6, 14, 18, 16, 0, 12}, {6, 14, 18, 16, 6, 12}, {6, 16, 0, 
            10, 18, 10}, {6, 16, 0, 14, 18, 10}, {6, 16, 0, 14, 18, 12}, {6, 
            16, 6, 8, 14, 18}, {6, 16, 6, 10, 20, 12}, {6, 16, 6, 12, 18, 
            14}, {6, 16, 6, 12, 20, 10}, {6, 16, 6, 14, 18, 12}, {6, 16, 6, 
            18, 14, 8}, {6, 16, 10, 6, 20, 12}, {6, 16, 10, 8, 18, 14}, {6, 
            16, 12, 6, 20, 10}, {6, 16, 18, 8, 10, 14}, {6, 16, 18, 12, 6, 
            14}, {6, 18, 0, 8, 18, 10}, {6, 18, 0, 8, 18, 12}, {6, 18, 0, 12, 
            18, 10}, {6, 18, 0, 12, 18, 12}, {6, 18, 8, 0, 14, 14}, {6, 18, 
            8, 0, 14, 16}, {6, 18, 8, 0, 18, 10}, {6, 18, 8, 0, 18, 12}, {6, 
            18, 8, 8, 14, 16}, {6, 18, 8, 10, 16, 14}, {6, 18, 10, 0, 12, 
            14}, {6, 18, 10, 0, 12, 16}, {6, 18, 10, 0, 16, 10}, {6, 18, 12, 
            0, 14, 16}, {6, 18, 12, 0, 18, 10}, {6, 18, 12, 0, 18, 12}, {6, 
            18, 12, 6, 16, 14}, {6, 18, 12, 8, 12, 16}, {6, 18, 12, 10, 14, 
            14}, {6, 18, 14, 0, 12, 14}, {6, 18, 14, 0, 12, 16}, {6, 18, 14, 
            0, 16, 10}, {6, 18, 14, 0, 16, 12}, {6, 18, 14, 6, 12, 16}, {6, 
            18, 14, 6, 16, 12}, {6, 18, 14, 8, 10, 16}, {6, 18, 14, 8, 14, 
            14}, {6, 18, 14, 10, 12, 14}, {6, 18, 14, 16, 6, 8}, {6, 18, 16, 
            6, 12, 14}, {6, 18, 16, 10, 8, 14}, {6, 20, 6, 8, 14, 14}, {6, 
            20, 6, 10, 16, 12}, {6, 20, 6, 12, 16, 10}, {6, 20, 6, 14, 14, 
            8}, {6, 20, 10, 6, 16, 12}, {6, 20, 12, 6, 16, 10}, {6, 22, 8, 8, 
            12, 12}, {6, 22, 8, 10, 14, 10}, {6, 22, 8, 12, 12, 8}, {6, 22, 
            10, 6, 12, 12}, {6, 22, 10, 8, 8, 12}, {6, 22, 10, 8, 10, 
            12}, {6, 22, 10, 8, 14, 10}, {6, 22, 10, 10, 12, 10}, {6, 22, 10, 
            12, 8, 6}, {6, 22, 10, 12, 8, 8}, {6, 22, 10, 12, 10, 8}, {6, 22, 
            12, 6, 12, 10}, {6, 22, 12, 8, 8, 8}, {6, 22, 12, 10, 8, 6}, {6, 
            22, 12, 10, 8, 10}, {8, 0, 10, 16, 10, 18}, {8, 0, 10, 16, 14, 
            18}, {8, 0, 10, 18, 10, 14}, {8, 0, 10, 18, 14, 14}, {8, 0, 12, 
            18, 14, 16}, {8, 0, 12, 20, 8, 12}, {8, 0, 12, 20, 8, 14}, {8, 0, 
            12, 20, 12, 12}, {8, 0, 12, 20, 12, 14}, {8, 0, 16, 18, 10, 
            8}, {8, 0, 16, 18, 14, 8}, {8, 0, 16, 18, 14, 12}, {8, 0, 16, 20, 
            8, 8}, {8, 0, 16, 20, 8, 10}, {8, 0, 16, 20, 12, 8}, {8, 0, 16, 
            20, 12, 10}, {8, 0, 18, 16, 10, 8}, {8, 0, 18, 16, 14, 8}, {8, 0, 
            18, 16, 14, 10}, {8, 0, 20, 16, 8, 8}, {8, 0, 20, 16, 8, 10}, {8, 
            0, 20, 16, 12, 8}, {8, 0, 20, 16, 12, 10}, {8, 6, 8, 10, 12, 
            22}, {8, 6, 8, 12, 8, 22}, {8, 6, 8, 22, 8, 12}, {8, 6, 8, 22, 
            12, 10}, {8, 6, 10, 12, 6, 22}, {8, 6, 10, 12, 10, 22}, {8, 6, 
            10, 16, 14, 18}, {8, 6, 10, 18, 16, 14}, {8, 6, 10, 22, 12, 
            10}, {8, 6, 12, 10, 6, 22}, {8, 6, 12, 10, 8, 22}, {8, 6, 12, 18, 
            16, 12}, {8, 6, 12, 22, 12, 8}, {8, 6, 14, 16, 18, 6}, {8, 6, 14, 
            18, 14, 14}, {8, 6, 14, 18, 16, 8}, {8, 6, 14, 20, 14, 6}, {8, 6, 
            14, 22, 10, 10}, {8, 6, 18, 14, 14, 0}, {8, 6, 18, 14, 16, 
            0}, {8, 6, 18, 14, 16, 8}, {8, 6, 18, 16, 14, 10}, {8, 6, 18, 18, 
            10, 0}, {8, 6, 18, 18, 12, 0}, {8, 6, 22, 12, 8, 12}, {8, 6, 22, 
            12, 12, 8}, {8, 6, 22, 14, 10, 10}, {8, 8, 6, 8, 12, 22}, {8, 8, 
            6, 8, 22, 12}, {8, 8, 6, 12, 10, 22}, {8, 8, 6, 12, 22, 10}, {8, 
            8, 8, 10, 12, 22}, {8, 8, 8, 10, 22, 12}, {8, 8, 8, 12, 10, 
            22}, {8, 8, 8, 12, 22, 10}, {8, 8, 8, 14, 16, 18}, {8, 8, 8, 14, 
            18, 16}, {8, 8, 8, 16, 14, 18}, {8, 8, 8, 16, 18, 14}, {8, 8, 8, 
            18, 14, 16}, {8, 8, 8, 18, 16, 14}, {8, 8, 8, 22, 10, 12}, {8, 8, 
            8, 22, 12, 10}, {8, 8, 10, 12, 6, 22}, {8, 8, 10, 12, 8, 22}, {8, 
            8, 10, 12, 22, 6}, {8, 8, 10, 12, 22, 8}, {8, 8, 10, 16, 12, 
            18}, {8, 8, 10, 16, 18, 12}, {8, 8, 12, 8, 6, 22}, {8, 8, 12, 8, 
            22, 6}, {8, 8, 12, 10, 8, 22}, {8, 8, 12, 10, 22, 8}, {8, 8, 12, 
            18, 10, 16}, {8, 8, 12, 18, 16, 10}, {8, 8, 12, 22, 6, 12}, {8, 
            8, 12, 22, 12, 6}, {8, 8, 16, 20, 0, 8}, {8, 8, 16, 20, 0, 
            10}, {8, 8, 16, 20, 8, 0}, {8, 8, 16, 20, 10, 0}, {8, 8, 20, 12, 
            0, 12}, {8, 8, 20, 12, 0, 14}, {8, 8, 20, 12, 12, 0}, {8, 8, 20, 
            12, 14, 0}, {8, 8, 20, 16, 0, 8}, {8, 8, 20, 16, 0, 10}, {8, 8, 
            20, 16, 8, 0}, {8, 8, 20, 16, 10, 0}, {8, 8, 22, 8, 6, 12}, {8, 
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            10, 14, 14}, {18, 8, 12, 6, 0, 18}, {18, 8, 14, 0, 8, 16}, {18, 
            8, 14, 0, 12, 16}, {18, 8, 14, 0, 14, 10}, {18, 8, 14, 0, 16, 
            12}, {18, 8, 14, 6, 14, 14}, {18, 8, 14, 8, 16, 8}, {18, 8, 14, 
            10, 14, 10}, {18, 8, 16, 6, 8, 14}, {18, 8, 16, 6, 12, 12}, {18, 
            8, 16, 6, 14, 10}, {18, 8, 16, 8, 10, 12}, {18, 8, 16, 8, 14, 
            8}, {18, 8, 16, 10, 10, 10}, {18, 8, 16, 10, 12, 8}, {18, 8, 16, 
            14, 6, 6}, {18, 10, 0, 8, 6, 18}, {18, 10, 0, 10, 6, 16}, {18, 
            10, 0, 12, 6, 18}, {18, 10, 0, 14, 6, 16}, {18, 10, 0, 14, 10, 
            16}, {18, 10, 6, 10, 0, 16}, {18, 10, 6, 14, 0, 12}, {18, 10, 6, 
            16, 0, 12}, {18, 10, 8, 0, 6, 18}, {18, 10, 8, 8, 0, 16}, {18, 
            10, 8, 14, 0, 10}, {18, 10, 8, 14, 10, 14}, {18, 10, 8, 16, 8, 
            12}, {18, 10, 10, 0, 16, 8}, {18, 10, 10, 8, 16, 10}, {18, 10, 
            10, 10, 14, 12}, {18, 10, 10, 14, 10, 12}, {18, 10, 10, 16, 0, 
            8}, {18, 10, 10, 16, 8, 10}, {18, 10, 12, 0, 6, 18}, {18, 10, 12, 
            8, 16, 8}, {18, 10, 12, 10, 14, 10}, {18, 10, 14, 0, 8, 16}, {18, 
            10, 14, 0, 10, 16}, {18, 10, 14, 0, 16, 8}, {18, 10, 14, 0, 16, 
            10}, {18, 10, 14, 6, 8, 16}, {18, 10, 14, 6, 12, 14}, {18, 10, 
            14, 6, 14, 12}, {18, 10, 14, 6, 16, 8}, {18, 10, 14, 8, 10, 
            14}, {18, 10, 14, 8, 14, 10}, {18, 10, 14, 10, 10, 12}, {18, 10, 
            14, 10, 12, 10}, {18, 12, 0, 8, 6, 18}, {18, 12, 0, 12, 6, 
            18}, {18, 12, 0, 14, 6, 16}, {18, 12, 0, 14, 8, 16}, {18, 12, 6, 
            10, 0, 18}, {18, 12, 6, 12, 0, 18}, {18, 12, 6, 14, 6, 16}, {18, 
            12, 6, 14, 10, 14}, {18, 12, 6, 16, 0, 14}, {18, 12, 6, 16, 8, 
            12}, {18, 12, 8, 0, 6, 18}, {18, 12, 10, 6, 0, 18}, {18, 12, 10, 
            14, 10, 10}, {18, 12, 10, 16, 8, 8}, {18, 12, 12, 0, 6, 18}, {18, 
            12, 12, 6, 0, 18}, {18, 14, 0, 8, 6, 14}, {18, 14, 0, 10, 6, 
            12}, {18, 14, 0, 10, 8, 10}, {18, 14, 0, 14, 6, 12}, {18, 14, 0, 
            14, 8, 10}, {18, 14, 6, 8, 16, 6}, {18, 14, 6, 10, 0, 16}, {18, 
            14, 6, 12, 0, 16}, {18, 14, 6, 12, 6, 16}, {18, 14, 6, 14, 0, 
            12}, {18, 14, 6, 14, 8, 14}, {18, 14, 6, 14, 10, 12}, {18, 14, 6, 
            16, 0, 12}, {18, 14, 6, 16, 6, 12}, {18, 14, 6, 16, 8, 10}, {18, 
            14, 8, 8, 0, 16}, {18, 14, 8, 12, 0, 16}, {18, 14, 8, 14, 0, 
            10}, {18, 14, 8, 14, 6, 14}, {18, 14, 8, 14, 10, 10}, {18, 14, 8, 
            16, 0, 12}, {18, 14, 8, 16, 8, 8}, {18, 14, 10, 8, 0, 16}, {18, 
            14, 10, 8, 6, 16}, {18, 14, 10, 10, 0, 16}, {18, 14, 10, 10, 8, 
            14}, {18, 14, 10, 10, 10, 12}, {18, 14, 10, 12, 6, 14}, {18, 14, 
            10, 12, 10, 10}, {18, 14, 10, 14, 6, 12}, {18, 14, 10, 14, 8, 
            10}, {18, 14, 10, 16, 0, 8}, {18, 14, 10, 16, 0, 10}, {18, 14, 
            10, 16, 6, 8}, {18, 16, 0, 8, 6, 14}, {18, 16, 0, 10, 6, 
            12}, {18, 16, 0, 10, 10, 8}, {18, 16, 0, 12, 6, 14}, {18, 16, 0, 
            14, 6, 12}, {18, 16, 0, 14, 8, 12}, {18, 16, 0, 14, 10, 8}, {18, 
            16, 0, 14, 10, 10}, {18, 16, 6, 14, 6, 12}, {18, 16, 6, 14, 10, 
            8}, {18, 16, 8, 6, 14, 6}, {18, 16, 8, 8, 6, 14}, {18, 16, 8, 10, 
            8, 12}, {18, 16, 8, 10, 10, 10}, {18, 16, 8, 12, 6, 12}, {18, 16, 
            8, 12, 10, 8}, {18, 16, 8, 14, 6, 10}, {18, 16, 8, 14, 8, 
            8}, {20, 0, 8, 8, 8, 16}, {20, 0, 8, 8, 12, 16}, {20, 0, 8, 10, 
            8, 16}, {20, 0, 8, 10, 12, 16}, {20, 0, 10, 8, 8, 16}, {20, 0, 
            10, 8, 12, 16}, {20, 0, 12, 8, 8, 12}, {20, 0, 12, 8, 12, 
            12}, {20, 0, 14, 8, 8, 12}, {20, 0, 14, 8, 12, 12}, {20, 0, 14, 
            10, 8, 10}, {20, 0, 14, 10, 12, 10}, {20, 6, 6, 8, 14, 14}, {20, 
            6, 6, 10, 12, 16}, {20, 6, 6, 12, 10, 16}, {20, 6, 6, 14, 8, 
            14}, {20, 6, 10, 6, 12, 16}, {20, 6, 12, 6, 10, 16}, {20, 8, 0, 
            8, 8, 16}, {20, 8, 0, 8, 10, 16}, {20, 8, 0, 12, 8, 16}, {20, 8, 
            0, 12, 10, 16}, {20, 8, 8, 0, 8, 16}, {20, 8, 8, 0, 10, 16}, {20, 
            8, 8, 0, 12, 12}, {20, 8, 8, 0, 14, 12}, {20, 8, 8, 8, 0, 
            16}, {20, 8, 8, 10, 0, 16}, {20, 8, 8, 12, 0, 12}, {20, 8, 8, 14, 
            0, 12}, {20, 8, 10, 8, 0, 16}, {20, 8, 10, 14, 0, 10}, {20, 8, 
            12, 0, 8, 16}, {20, 8, 12, 0, 10, 16}, {20, 8, 12, 0, 12, 
            12}, {20, 8, 12, 0, 14, 12}, {20, 8, 14, 6, 6, 14}, {20, 10, 0, 
            8, 8, 16}, {20, 10, 0, 12, 8, 16}, {20, 10, 6, 12, 6, 16}, {20, 
            10, 8, 0, 8, 16}, {20, 10, 8, 0, 14, 10}, {20, 10, 12, 0, 8, 
            16}, {20, 10, 12, 0, 14, 10}, {20, 10, 12, 6, 6, 16}, {20, 12, 0, 
            8, 8, 12}, {20, 12, 0, 12, 8, 12}, {20, 12, 6, 10, 6, 16}, {20, 
            12, 8, 8, 0, 16}, {20, 12, 8, 10, 0, 16}, {20, 12, 8, 12, 0, 
            12}, {20, 12, 8, 14, 0, 12}, {20, 12, 10, 6, 6, 16}, {20, 12, 10, 
            8, 0, 16}, {20, 12, 10, 14, 0, 10}, {20, 14, 0, 8, 8, 12}, {20, 
            14, 0, 8, 10, 10}, {20, 14, 0, 12, 8, 12}, {20, 14, 0, 12, 10, 
            10}, {20, 14, 8, 6, 6, 14}, {22, 6, 8, 8, 12, 12}, {22, 6, 8, 10, 
            10, 14}, {22, 6, 8, 12, 8, 12}, {22, 6, 10, 6, 12, 12}, {22, 6, 
            10, 8, 10, 14}, {22, 6, 10, 8, 12, 8}, {22, 6, 10, 8, 12, 
            10}, {22, 6, 10, 10, 10, 12}, {22, 6, 10, 12, 6, 8}, {22, 6, 10, 
            12, 8, 8}, {22, 6, 10, 12, 8, 10}, {22, 6, 12, 6, 10, 12}, {22, 
            6, 12, 8, 8, 8}, {22, 6, 12, 10, 6, 8}, {22, 6, 12, 10, 10, 
            8}, {22, 8, 6, 8, 12, 12}, {22, 8, 6, 10, 10, 14}, {22, 8, 6, 12, 
            8, 12}, {22, 8, 8, 6, 12, 8}, {22, 8, 8, 12, 6, 8}, {22, 8, 10, 
            6, 10, 14}, {22, 8, 10, 8, 12, 8}, {22, 8, 10, 10, 10, 10}, {22, 
            8, 10, 12, 8, 8}, {22, 8, 12, 6, 8, 12}, {22, 8, 12, 6, 10, 
            8}, {22, 8, 12, 6, 10, 10}, {22, 8, 12, 8, 6, 12}, {22, 8, 12, 8, 
            10, 8}, {22, 8, 12, 10, 6, 8}, {22, 8, 12, 10, 6, 10}, {22, 8, 
            12, 10, 8, 8}, {22, 10, 6, 6, 12, 8}, {22, 10, 6, 8, 12, 8}, {22, 
            10, 6, 8, 12, 10}, {22, 10, 6, 10, 8, 14}, {22, 10, 6, 10, 10, 
            12}, {22, 10, 6, 12, 6, 12}, {22, 10, 6, 12, 8, 8}, {22, 10, 6, 
            12, 8, 10}, {22, 10, 8, 8, 12, 8}, {22, 10, 8, 10, 6, 14}, {22, 
            10, 8, 10, 10, 10}, {22, 10, 8, 12, 8, 8}, {22, 10, 10, 6, 8, 
            14}, {22, 10, 10, 6, 10, 12}, {22, 10, 10, 6, 12, 8}, {22, 10, 
            10, 8, 6, 14}, {22, 10, 10, 8, 10, 10}, {22, 10, 10, 10, 6, 
            12}, {22, 10, 10, 10, 8, 10}, {22, 10, 10, 12, 6, 8}, {22, 12, 6, 
            6, 10, 8}, {22, 12, 6, 8, 8, 8}, {22, 12, 6, 10, 6, 12}, {22, 12, 
            6, 10, 10, 8}, {22, 12, 8, 6, 8, 12}, {22, 12, 8, 6, 10, 8}, {22, 
            12, 8, 6, 10, 10}, {22, 12, 8, 8, 6, 12}, {22, 12, 8, 8, 10, 
            8}, {22, 12, 8, 10, 6, 8}, {22, 12, 8, 10, 6, 10}, {22, 12, 8, 
            10, 8, 8}, {4, 6, 8, 14, 16, 18}, {4, 6, 8, 16, 12, 18}, {4, 6, 
            8, 16, 18, 10}, {4, 6, 8, 18, 14, 10}, {4, 6, 10, 16, 14, 
            18}, {4, 6, 12, 22, 8, 12}, {4, 6, 12, 22, 10, 6}, {4, 6, 12, 22, 
            10, 12}, {4, 6, 12, 22, 12, 6}, {4, 6, 14, 16, 6, 18}, {4, 6, 14, 
            16, 10, 18}, {4, 6, 14, 16, 18, 10}, {4, 6, 14, 18, 12, 16}, {4, 
            6, 14, 18, 16, 12}, {4, 6, 14, 20, 10, 14}, {4, 6, 14, 20, 14, 
            10}, {4, 6, 14, 22, 6, 8}, {4, 6, 14, 22, 6, 10}, {4, 6, 16, 14, 
            6, 18}, {4, 6, 16, 14, 8, 18}, {4, 6, 16, 14, 18, 8}, {4, 6, 18, 
            18, 12, 12}, {4, 6, 20, 14, 14, 8}, {4, 6, 22, 14, 6, 8}, {4, 6, 
            22, 14, 6, 10}, {4, 8, 6, 12, 16, 18}, {4, 8, 6, 14, 18, 10}, {4, 
            8, 6, 16, 14, 18}, {4, 8, 6, 18, 16, 10}, {4, 8, 8, 14, 16, 
            18}, {4, 8, 8, 16, 14, 18}, {4, 8, 10, 12, 22, 8}, {4, 8, 10, 12, 
            22, 10}, {4, 8, 10, 22, 10, 12}, {4, 8, 10, 22, 12, 6}, {4, 8, 
            12, 18, 16, 14}, {4, 8, 12, 22, 8, 12}, {4, 8, 12, 22, 12, 
            10}, {4, 8, 14, 16, 6, 18}, {4, 8, 14, 16, 8, 18}, {4, 8, 14, 18, 
            10, 16}, {4, 8, 14, 20, 8, 14}, {4, 8, 16, 14, 8, 18}, {4, 8, 16, 
            18, 12, 14}, {4, 8, 16, 20, 10, 12}, {4, 8, 20, 14, 8, 14}, {4, 
            8, 20, 16, 10, 12}, {4, 8, 22, 12, 6, 12}, {4, 8, 22, 12, 8, 
            12}, {4, 8, 22, 12, 12, 8}, {4, 10, 6, 14, 16, 18}, {4, 10, 8, 
            10, 22, 12}, {4, 10, 8, 12, 22, 6}, {4, 10, 8, 22, 12, 8}, {4, 
            10, 8, 22, 12, 10}, {4, 10, 10, 12, 22, 10}, {4, 10, 10, 16, 18, 
            14}, {4, 10, 10, 18, 16, 14}, {4, 10, 10, 22, 12, 10}, {4, 10, 
            12, 10, 22, 8}, {4, 10, 16, 14, 6, 18}, {4, 10, 16, 18, 10, 
            14}, {4, 10, 16, 20, 8, 12}, {4, 10, 18, 14, 8, 16}, {4, 10, 18, 
            16, 10, 14}, {4, 10, 20, 14, 6, 14}, {4, 10, 20, 16, 8, 12}, {4, 
            10, 22, 10, 8, 12}, {4, 10, 22, 10, 12, 8}, {4, 10, 22, 12, 6, 
            6}, {4, 10, 22, 12, 6, 12}, {4, 12, 6, 8, 22, 12}, {4, 12, 6, 10, 
            22, 6}, {4, 12, 6, 10, 22, 12}, {4, 12, 6, 12, 22, 6}, {4, 12, 8, 
            8, 22, 12}, {4, 12, 8, 12, 22, 10}, {4, 12, 8, 16, 18, 14}, {4, 
            12, 10, 22, 10, 8}, {4, 12, 12, 8, 22, 8}, {4, 12, 12, 22, 8, 
            8}, {4, 12, 16, 8, 6, 18}, {4, 12, 18, 14, 6, 16}, {4, 12, 18, 
            16, 8, 14}, {4, 12, 18, 18, 6, 12}, {4, 12, 22, 8, 10, 8}, {4, 
            12, 22, 8, 10, 10}, {4, 12, 22, 10, 8, 6}, {4, 12, 22, 10, 10, 
            10}, {4, 12, 22, 12, 6, 6}, {4, 12, 22, 12, 8, 10}, {4, 14, 6, 6, 
            16, 18}, {4, 14, 6, 6, 22, 8}, {4, 14, 6, 6, 22, 10}, {4, 14, 6, 
            10, 16, 18}, {4, 14, 6, 10, 20, 14}, {4, 14, 6, 12, 18, 16}, {4, 
            14, 6, 14, 20, 10}, {4, 14, 6, 16, 18, 12}, {4, 14, 6, 18, 16, 
            10}, {4, 14, 8, 6, 16, 18}, {4, 14, 8, 8, 16, 18}, {4, 14, 8, 8, 
            20, 14}, {4, 14, 8, 10, 18, 16}, {4, 14, 14, 6, 20, 8}, {4, 14, 
            14, 20, 6, 8}, {4, 14, 16, 6, 8, 18}, {4, 14, 16, 8, 8, 18}, {4, 
            14, 16, 10, 6, 18}, {4, 14, 18, 6, 16, 8}, {4, 14, 18, 8, 6, 
            10}, {4, 14, 20, 14, 6, 10}, {4, 16, 6, 6, 14, 18}, {4, 16, 6, 8, 
            14, 18}, {4, 16, 6, 18, 14, 8}, {4, 16, 8, 8, 14, 18}, {4, 16, 8, 
            10, 20, 12}, {4, 16, 8, 12, 18, 14}, {4, 16, 10, 6, 14, 18}, {4, 
            16, 10, 8, 20, 12}, {4, 16, 10, 10, 18, 14}, {4, 16, 12, 6, 8, 
            18}, {4, 16, 14, 6, 10, 18}, {4, 16, 14, 8, 6, 18}, {4, 16, 14, 
            8, 8, 18}, {4, 16, 18, 6, 8, 10}, {4, 16, 18, 6, 14, 10}, {4, 16, 
            18, 10, 10, 14}, {4, 16, 18, 12, 8, 14}, {4, 16, 18, 14, 6, 
            12}, {4, 18, 6, 12, 18, 12}, {4, 18, 10, 8, 14, 16}, {4, 18, 10, 
            10, 16, 14}, {4, 18, 12, 6, 14, 16}, {4, 18, 12, 6, 18, 12}, {4, 
            18, 12, 8, 16, 14}, {4, 18, 14, 6, 8, 10}, {4, 18, 14, 16, 6, 
            8}, {4, 18, 16, 6, 14, 12}, {4, 18, 16, 8, 6, 10}, {4, 18, 16, 8, 
            12, 14}, {4, 18, 16, 10, 10, 14}, {4, 18, 16, 14, 6, 10}, {4, 20, 
            6, 14, 14, 8}, {4, 20, 8, 8, 14, 14}, {4, 20, 8, 10, 16, 12}, {4, 
            20, 10, 6, 14, 14}, {4, 20, 10, 8, 16, 12}, {4, 20, 14, 6, 14, 
            10}, {4, 22, 6, 6, 14, 8}, {4, 22, 6, 6, 14, 10}, {4, 22, 8, 6, 
            12, 12}, {4, 22, 8, 8, 12, 12}, {4, 22, 8, 12, 12, 8}, {4, 22, 
            10, 6, 12, 6}, {4, 22, 10, 6, 12, 12}, {4, 22, 10, 8, 10, 
            12}, {4, 22, 10, 12, 10, 8}, {4, 22, 12, 6, 12, 6}, {4, 22, 12, 
            8, 10, 6}, {4, 22, 12, 8, 12, 10}, {4, 22, 12, 10, 8, 8}, {4, 22, 
            12, 10, 8, 10}, {4, 22, 12, 10, 10, 10}, {6, 4, 8, 14, 18, 
            16}, {6, 4, 8, 16, 10, 18}, {6, 4, 8, 16, 18, 12}, {6, 4, 8, 18, 
            10, 14}, {6, 4, 10, 16, 18, 14}, {6, 4, 12, 22, 6, 10}, {6, 4, 
            12, 22, 6, 12}, {6, 4, 12, 22, 12, 8}, {6, 4, 12, 22, 12, 
            10}, {6, 4, 14, 16, 10, 18}, {6, 4, 14, 16, 18, 6}, {6, 4, 14, 
            16, 18, 10}, {6, 4, 14, 18, 12, 16}, {6, 4, 14, 18, 16, 12}, {6, 
            4, 14, 20, 10, 14}, {6, 4, 14, 20, 14, 10}, {6, 4, 14, 22, 8, 
            6}, {6, 4, 14, 22, 10, 6}, {6, 4, 16, 14, 8, 18}, {6, 4, 16, 14, 
            18, 6}, {6, 4, 16, 14, 18, 8}, {6, 4, 18, 18, 12, 12}, {6, 4, 20, 
            14, 8, 14}, {6, 4, 22, 14, 8, 6}, {6, 4, 22, 14, 10, 6}, {6, 6, 
            22, 12, 4, 10}, {6, 6, 22, 12, 4, 12}, {6, 6, 22, 12, 10, 4}, {6, 
            6, 22, 12, 12, 4}, {6, 8, 4, 10, 16, 18}, {6, 8, 4, 10, 18, 
            14}, {6, 8, 4, 18, 14, 16}, {6, 8, 4, 18, 16, 12}, {6, 8, 14, 16, 
            4, 18}, {6, 8, 14, 20, 4, 14}, {6, 8, 14, 22, 4, 6}, {6, 8, 22, 
            10, 12, 4}, {6, 8, 22, 14, 4, 6}, {6, 10, 4, 18, 16, 14}, {6, 10, 
            12, 8, 22, 4}, {6, 10, 12, 22, 6, 4}, {6, 10, 14, 22, 4, 6}, {6, 
            10, 16, 8, 4, 18}, {6, 10, 16, 14, 4, 18}, {6, 10, 18, 8, 4, 
            14}, {6, 10, 20, 14, 4, 14}, {6, 10, 22, 14, 4, 6}, {6, 12, 4, 6, 
            22, 10}, {6, 12, 4, 6, 22, 12}, {6, 12, 4, 12, 22, 8}, {6, 12, 4, 
            12, 22, 10}, {6, 12, 10, 6, 22, 4}, {6, 12, 10, 22, 8, 4}, {6, 
            12, 12, 6, 22, 4}, {6, 12, 12, 22, 6, 4}, {6, 12, 18, 14, 4, 
            16}, {6, 12, 18, 18, 4, 12}, {6, 12, 22, 12, 4, 8}, {6, 12, 22, 
            12, 4, 10}, {6, 14, 4, 8, 22, 6}, {6, 14, 4, 10, 16, 18}, {6, 14, 
            4, 10, 20, 14}, {6, 14, 4, 10, 22, 6}, {6, 14, 4, 12, 18, 
            16}, {6, 14, 4, 14, 20, 10}, {6, 14, 4, 16, 18, 12}, {6, 14, 4, 
            18, 16, 6}, {6, 14, 4, 18, 16, 10}, {6, 14, 8, 4, 16, 18}, {6, 
            14, 8, 4, 20, 14}, {6, 14, 8, 4, 22, 6}, {6, 14, 10, 4, 22, 
            6}, {6, 14, 18, 4, 8, 16}, {6, 14, 18, 4, 16, 6}, {6, 14, 18, 4, 
            16, 8}, {6, 14, 20, 14, 4, 10}, {6, 16, 4, 8, 14, 18}, {6, 16, 4, 
            18, 14, 6}, {6, 16, 4, 18, 14, 8}, {6, 16, 10, 4, 8, 18}, {6, 16, 
            10, 4, 14, 18}, {6, 16, 18, 4, 8, 12}, {6, 16, 18, 4, 10, 
            14}, {6, 16, 18, 4, 14, 6}, {6, 16, 18, 4, 14, 10}, {6, 16, 18, 
            14, 4, 12}, {6, 18, 4, 12, 18, 12}, {6, 18, 10, 4, 8, 14}, {6, 
            18, 12, 4, 14, 16}, {6, 18, 12, 4, 18, 12}, {6, 18, 14, 8, 4, 
            16}, {6, 18, 14, 16, 4, 6}, {6, 18, 14, 16, 4, 8}, {6, 18, 16, 4, 
            14, 12}, {6, 18, 16, 8, 4, 12}, {6, 18, 16, 10, 4, 14}, {6, 18, 
            16, 14, 4, 6}, {6, 18, 16, 14, 4, 10}, {6, 20, 4, 8, 14, 14}, {6, 
            20, 10, 4, 14, 14}, {6, 20, 14, 4, 14, 10}, {6, 22, 4, 8, 14, 
            6}, {6, 22, 4, 10, 14, 6}, {6, 22, 6, 4, 12, 10}, {6, 22, 6, 4, 
            12, 12}, {6, 22, 6, 10, 12, 4}, {6, 22, 6, 12, 12, 4}, {6, 22, 8, 
            4, 14, 6}, {6, 22, 8, 12, 10, 4}, {6, 22, 10, 4, 14, 6}, {6, 22, 
            12, 4, 12, 8}, {6, 22, 12, 4, 12, 10}, {8, 4, 6, 12, 18, 16}, {8, 
            4, 6, 14, 10, 18}, {8, 4, 6, 16, 18, 14}, {8, 4, 6, 18, 10, 
            16}, {8, 4, 8, 14, 18, 16}, {8, 4, 8, 16, 18, 14}, {8, 4, 10, 12, 
            8, 22}, {8, 4, 10, 12, 10, 22}, {8, 4, 10, 22, 6, 12}, {8, 4, 10, 
            22, 12, 10}, {8, 4, 12, 18, 14, 16}, {8, 4, 12, 22, 10, 12}, {8, 
            4, 12, 22, 12, 8}, {8, 4, 14, 16, 18, 6}, {8, 4, 14, 16, 18, 
            8}, {8, 4, 14, 18, 16, 10}, {8, 4, 14, 20, 14, 8}, {8, 4, 16, 14, 
            18, 8}, {8, 4, 16, 18, 14, 12}, {8, 4, 16, 20, 12, 10}, {8, 4, 
            20, 14, 14, 8}, {8, 4, 20, 16, 12, 10}, {8, 4, 22, 12, 8, 
            12}, {8, 4, 22, 12, 12, 6}, {8, 4, 22, 12, 12, 8}, {8, 6, 4, 10, 
            14, 18}, {8, 6, 4, 10, 18, 16}, {8, 6, 4, 18, 12, 16}, {8, 6, 4, 
            18, 16, 14}, {8, 6, 14, 16, 18, 4}, {8, 6, 14, 20, 14, 4}, {8, 6, 
            14, 22, 6, 4}, {8, 6, 22, 10, 4, 12}, {8, 6, 22, 14, 6, 4}, {8, 
            8, 4, 18, 14, 16}, {8, 8, 4, 18, 16, 14}, {8, 8, 12, 10, 4, 
            22}, {8, 8, 12, 10, 22, 4}, {8, 8, 12, 22, 4, 12}, {8, 8, 12, 22, 
            12, 4}, {8, 10, 4, 6, 22, 12}, {8, 10, 4, 8, 12, 22}, {8, 10, 4, 
            10, 12, 22}, {8, 10, 4, 12, 22, 10}, {8, 10, 10, 12, 22, 4}, {8, 
            10, 10, 22, 12, 4}, {8, 10, 12, 10, 4, 22}, {8, 10, 14, 6, 4, 
            18}, {8, 10, 18, 6, 4, 16}, {8, 10, 22, 8, 12, 4}, {8, 10, 22, 
            12, 4, 12}, {8, 12, 4, 10, 22, 12}, {8, 12, 4, 12, 22, 8}, {8, 
            12, 4, 14, 18, 16}, {8, 12, 8, 4, 10, 22}, {8, 12, 8, 4, 22, 
            12}, {8, 12, 8, 12, 22, 4}, {8, 12, 8, 22, 10, 4}, {8, 12, 10, 4, 
            10, 22}, {8, 12, 12, 4, 22, 6}, {8, 12, 12, 4, 22, 8}, {8, 12, 
            12, 22, 4, 6}, {8, 12, 12, 22, 4, 8}, {8, 12, 16, 20, 4, 10}, {8, 
            12, 18, 4, 6, 16}, {8, 12, 20, 16, 4, 10}, {8, 12, 22, 10, 4, 
            10}, {8, 12, 22, 10, 10, 4}, {8, 12, 22, 12, 4, 8}, {8, 12, 22, 
            12, 8, 4}, {8, 14, 4, 14, 20, 8}, {8, 14, 4, 16, 18, 10}, {8, 14, 
            4, 18, 16, 6}, {8, 14, 4, 18, 16, 8}, {8, 14, 6, 6, 22, 4}, {8, 
            14, 6, 14, 20, 4}, {8, 14, 6, 18, 16, 4}, {8, 14, 10, 4, 6, 
            18}, {8, 14, 14, 4, 20, 8}, {8, 14, 14, 20, 4, 8}, {8, 14, 18, 4, 
            8, 16}, {8, 14, 18, 4, 16, 8}, {8, 14, 18, 12, 4, 16}, {8, 14, 
            18, 16, 4, 12}, {8, 14, 20, 14, 4, 8}, {8, 14, 20, 14, 6, 4}, {8, 
            16, 4, 12, 20, 10}, {8, 16, 4, 14, 18, 12}, {8, 16, 4, 18, 14, 
            8}, {8, 16, 12, 4, 20, 10}, {8, 16, 18, 4, 6, 14}, {8, 16, 18, 4, 
            8, 14}, {8, 16, 18, 4, 14, 6}, {8, 16, 18, 4, 14, 8}, {8, 16, 18, 
            6, 14, 4}, {8, 16, 18, 14, 4, 10}, {8, 18, 10, 4, 6, 16}, {8, 18, 
            12, 6, 4, 16}, {8, 18, 14, 4, 12, 16}, {8, 18, 14, 4, 16, 
            12}, {8, 18, 14, 8, 4, 16}, {8, 18, 14, 16, 4, 8}, {8, 18, 16, 4, 
            14, 10}, {8, 18, 16, 6, 4, 14}, {8, 18, 16, 8, 4, 14}, {8, 18, 
            16, 14, 4, 6}, {8, 18, 16, 14, 4, 8}, {8, 18, 16, 14, 6, 4}, {8, 
            20, 4, 12, 16, 10}, {8, 20, 4, 14, 14, 8}, {8, 20, 12, 4, 16, 
            10}, {8, 20, 14, 4, 14, 8}, {8, 20, 14, 6, 14, 4}, {8, 22, 4, 8, 
            12, 12}, {8, 22, 4, 12, 12, 6}, {8, 22, 4, 12, 12, 8}, {8, 22, 6, 
            4, 10, 12}, {8, 22, 6, 6, 14, 4}, {8, 22, 10, 4, 12, 12}, {8, 22, 
            10, 12, 8, 4}, {8, 22, 12, 4, 10, 10}, {8, 22, 12, 4, 12, 8}, {8, 
            22, 12, 8, 12, 4}, {8, 22, 12, 10, 10, 4}, {10, 4, 6, 14, 18, 
            16}, {10, 4, 8, 10, 12, 22}, {10, 4, 8, 12, 6, 22}, {10, 4, 8, 
            22, 8, 12}, {10, 4, 8, 22, 10, 12}, {10, 4, 10, 12, 10, 22}, {10, 
            4, 10, 16, 14, 18}, {10, 4, 10, 18, 14, 16}, {10, 4, 10, 22, 10, 
            12}, {10, 4, 12, 10, 8, 22}, {10, 4, 16, 14, 18, 6}, {10, 4, 16, 
            18, 14, 10}, {10, 4, 16, 20, 12, 8}, {10, 4, 18, 14, 16, 8}, {10, 
            4, 18, 16, 14, 10}, {10, 4, 20, 14, 14, 6}, {10, 4, 20, 16, 12, 
            8}, {10, 4, 22, 10, 8, 12}, {10, 4, 22, 10, 12, 8}, {10, 4, 22, 
            12, 6, 6}, {10, 4, 22, 12, 12, 6}, {10, 6, 4, 18, 14, 16}, {10, 
            6, 12, 8, 4, 22}, {10, 6, 12, 22, 4, 6}, {10, 6, 14, 22, 6, 
            4}, {10, 6, 16, 8, 18, 4}, {10, 6, 16, 14, 18, 4}, {10, 6, 18, 8, 
            14, 4}, {10, 6, 20, 14, 14, 4}, {10, 6, 22, 14, 6, 4}, {10, 8, 4, 
            6, 12, 22}, {10, 8, 4, 8, 22, 12}, {10, 8, 4, 10, 22, 12}, {10, 
            8, 4, 12, 10, 22}, {10, 8, 10, 12, 4, 22}, {10, 8, 10, 22, 4, 
            12}, {10, 8, 12, 10, 22, 4}, {10, 8, 14, 6, 18, 4}, {10, 8, 18, 
            6, 16, 4}, {10, 8, 22, 8, 4, 12}, {10, 8, 22, 12, 12, 4}, {10, 
            10, 4, 10, 12, 22}, {10, 10, 4, 10, 22, 12}, {10, 10, 4, 14, 16, 
            18}, {10, 10, 4, 14, 18, 16}, {10, 10, 8, 4, 12, 22}, {10, 10, 8, 
            4, 22, 12}, {10, 10, 12, 4, 8, 22}, {10, 10, 12, 4, 22, 8}, {10, 
            10, 12, 10, 4, 22}, {10, 10, 12, 10, 22, 4}, {10, 10, 22, 8, 4, 
            12}, {10, 10, 22, 8, 12, 4}, {10, 10, 22, 10, 4, 12}, {10, 10, 
            22, 10, 12, 4}, {10, 12, 4, 8, 10, 22}, {10, 12, 6, 4, 8, 
            22}, {10, 12, 6, 4, 22, 6}, {10, 12, 8, 22, 10, 4}, {10, 12, 10, 
            4, 10, 22}, {10, 12, 10, 8, 4, 22}, {10, 12, 10, 22, 4, 8}, {10, 
            12, 10, 22, 10, 4}, {10, 12, 12, 4, 22, 6}, {10, 12, 12, 8, 22, 
            4}, {10, 12, 12, 22, 4, 6}, {10, 12, 12, 22, 8, 4}, {10, 12, 16, 
            20, 4, 8}, {10, 12, 20, 16, 4, 8}, {10, 14, 6, 6, 22, 4}, {10, 
            14, 8, 18, 6, 4}, {10, 14, 14, 4, 20, 6}, {10, 14, 14, 6, 20, 
            4}, {10, 14, 14, 20, 4, 6}, {10, 14, 14, 20, 6, 4}, {10, 14, 16, 
            4, 18, 8}, {10, 14, 16, 10, 4, 18}, {10, 14, 16, 18, 4, 10}, {10, 
            14, 18, 4, 6, 16}, {10, 14, 18, 4, 16, 6}, {10, 14, 18, 6, 16, 
            4}, {10, 14, 18, 10, 4, 16}, {10, 14, 18, 16, 4, 10}, {10, 16, 4, 
            12, 20, 8}, {10, 16, 4, 14, 18, 10}, {10, 16, 4, 18, 14, 6}, {10, 
            16, 6, 18, 8, 4}, {10, 16, 6, 18, 14, 4}, {10, 16, 12, 4, 20, 
            8}, {10, 16, 14, 4, 10, 18}, {10, 16, 14, 4, 18, 10}, {10, 16, 
            14, 18, 4, 8}, {10, 18, 4, 14, 16, 10}, {10, 18, 4, 16, 14, 
            8}, {10, 18, 6, 14, 8, 4}, {10, 18, 8, 16, 6, 4}, {10, 18, 14, 4, 
            10, 16}, {10, 18, 14, 4, 16, 10}, {10, 18, 14, 6, 4, 16}, {10, 
            18, 14, 16, 4, 6}, {10, 18, 14, 16, 6, 4}, {10, 20, 4, 12, 16, 
            8}, {10, 20, 4, 14, 14, 6}, {10, 20, 6, 14, 14, 4}, {10, 20, 12, 
            4, 16, 8}, {10, 22, 4, 6, 12, 6}, {10, 22, 4, 8, 10, 12}, {10, 
            22, 4, 12, 10, 8}, {10, 22, 4, 12, 12, 6}, {10, 22, 6, 6, 14, 
            4}, {10, 22, 8, 4, 8, 12}, {10, 22, 8, 12, 12, 4}, {10, 22, 10, 
            4, 8, 12}, {10, 22, 10, 4, 10, 12}, {10, 22, 10, 12, 8, 4}, {10, 
            22, 10, 12, 10, 4}, {12, 4, 6, 8, 12, 22}, {12, 4, 6, 10, 6, 
            22}, {12, 4, 6, 10, 12, 22}, {12, 4, 6, 12, 6, 22}, {12, 4, 8, 8, 
            12, 22}, {12, 4, 8, 12, 10, 22}, {12, 4, 8, 16, 14, 18}, {12, 4, 
            10, 22, 8, 10}, {12, 4, 12, 8, 8, 22}, {12, 4, 12, 22, 8, 
            8}, {12, 4, 16, 8, 18, 6}, {12, 4, 18, 14, 16, 6}, {12, 4, 18, 
            16, 14, 8}, {12, 4, 18, 18, 12, 6}, {12, 4, 22, 8, 8, 10}, {12, 
            4, 22, 8, 10, 10}, {12, 4, 22, 10, 6, 8}, {12, 4, 22, 10, 10, 
            10}, {12, 4, 22, 12, 6, 6}, {12, 4, 22, 12, 10, 8}, {12, 6, 4, 6, 
            10, 22}, {12, 6, 4, 6, 12, 22}, {12, 6, 4, 12, 8, 22}, {12, 6, 4, 
            12, 10, 22}, {12, 6, 10, 6, 4, 22}, {12, 6, 10, 22, 4, 8}, {12, 
            6, 12, 6, 4, 22}, {12, 6, 12, 22, 4, 6}, {12, 6, 18, 14, 16, 
            4}, {12, 6, 18, 18, 12, 4}, {12, 6, 22, 12, 8, 4}, {12, 6, 22, 
            12, 10, 4}, {12, 8, 4, 10, 12, 22}, {12, 8, 4, 12, 8, 22}, {12, 
            8, 4, 14, 16, 18}, {12, 8, 8, 4, 12, 22}, {12, 8, 8, 4, 22, 
            10}, {12, 8, 8, 12, 4, 22}, {12, 8, 8, 22, 4, 10}, {12, 8, 10, 4, 
            22, 10}, {12, 8, 12, 4, 6, 22}, {12, 8, 12, 4, 8, 22}, {12, 8, 
            12, 22, 6, 4}, {12, 8, 12, 22, 8, 4}, {12, 8, 16, 20, 10, 
            4}, {12, 8, 18, 4, 16, 6}, {12, 8, 20, 16, 10, 4}, {12, 8, 22, 
            10, 4, 10}, {12, 8, 22, 10, 10, 4}, {12, 8, 22, 12, 4, 8}, {12, 
            8, 22, 12, 8, 4}, {12, 10, 4, 8, 22, 10}, {12, 10, 6, 4, 6, 
            22}, {12, 10, 6, 4, 22, 8}, {12, 10, 8, 22, 4, 10}, {12, 10, 10, 
            4, 22, 10}, {12, 10, 10, 8, 22, 4}, {12, 10, 10, 22, 4, 10}, {12, 
            10, 10, 22, 8, 4}, {12, 10, 12, 4, 6, 22}, {12, 10, 12, 8, 4, 
            22}, {12, 10, 12, 22, 4, 8}, {12, 10, 12, 22, 6, 4}, {12, 10, 16, 
            20, 8, 4}, {12, 10, 20, 16, 8, 4}, {12, 12, 4, 8, 8, 22}, {12, 
            12, 4, 8, 22, 8}, {12, 12, 6, 4, 6, 22}, {12, 12, 6, 4, 22, 
            6}, {12, 12, 8, 6, 4, 22}, {12, 12, 8, 6, 22, 4}, {12, 12, 8, 8, 
            4, 22}, {12, 12, 8, 8, 22, 4}, {12, 12, 10, 4, 8, 22}, {12, 12, 
            10, 4, 22, 8}, {12, 12, 10, 6, 4, 22}, {12, 12, 10, 6, 22, 
            4}, {12, 12, 18, 18, 4, 6}, {12, 12, 18, 18, 6, 4}, {12, 14, 16, 
            4, 18, 6}, {12, 14, 16, 6, 18, 4}, {12, 14, 16, 8, 4, 18}, {12, 
            14, 16, 18, 4, 8}, {12, 16, 4, 18, 8, 6}, {12, 16, 8, 10, 20, 
            4}, {12, 16, 10, 8, 20, 4}, {12, 16, 14, 4, 8, 18}, {12, 16, 14, 
            4, 18, 8}, {12, 16, 14, 18, 4, 6}, {12, 16, 14, 18, 6, 4}, {12, 
            18, 4, 12, 18, 6}, {12, 18, 4, 14, 16, 8}, {12, 18, 4, 16, 14, 
            6}, {12, 18, 6, 12, 18, 4}, {12, 18, 6, 16, 14, 4}, {12, 18, 8, 
            16, 4, 6}, {12, 18, 12, 4, 18, 6}, {12, 18, 12, 6, 18, 4}, {12, 
            20, 8, 10, 16, 4}, {12, 20, 10, 8, 16, 4}, {12, 22, 4, 6, 10, 
            8}, {12, 22, 4, 6, 12, 6}, {12, 22, 4, 8, 8, 10}, {12, 22, 4, 10, 
            8, 10}, {12, 22, 4, 10, 10, 10}, {12, 22, 4, 10, 12, 8}, {12, 22, 
            6, 8, 12, 4}, {12, 22, 6, 10, 12, 4}, {12, 22, 8, 4, 10, 
            10}, {12, 22, 8, 4, 12, 8}, {12, 22, 8, 8, 12, 4}, {12, 22, 8, 
            10, 10, 4}, {14, 4, 6, 6, 8, 22}, {14, 4, 6, 6, 10, 22}, {14, 4, 
            6, 6, 18, 16}, {14, 4, 6, 10, 14, 20}, {14, 4, 6, 10, 18, 
            16}, {14, 4, 6, 12, 16, 18}, {14, 4, 6, 14, 10, 20}, {14, 4, 6, 
            16, 12, 18}, {14, 4, 6, 18, 10, 16}, {14, 4, 8, 6, 18, 16}, {14, 
            4, 8, 8, 14, 20}, {14, 4, 8, 8, 18, 16}, {14, 4, 8, 10, 16, 
            18}, {14, 4, 14, 6, 8, 20}, {14, 4, 14, 20, 8, 6}, {14, 4, 16, 6, 
            18, 8}, {14, 4, 16, 8, 18, 8}, {14, 4, 16, 10, 18, 6}, {14, 4, 
            18, 6, 8, 16}, {14, 4, 18, 8, 10, 6}, {14, 4, 20, 14, 10, 
            6}, {14, 6, 4, 8, 6, 22}, {14, 6, 4, 10, 6, 22}, {14, 6, 4, 10, 
            14, 20}, {14, 6, 4, 10, 18, 16}, {14, 6, 4, 12, 16, 18}, {14, 6, 
            4, 14, 10, 20}, {14, 6, 4, 16, 12, 18}, {14, 6, 4, 18, 6, 
            16}, {14, 6, 4, 18, 10, 16}, {14, 6, 8, 4, 6, 22}, {14, 6, 8, 4, 
            14, 20}, {14, 6, 8, 4, 18, 16}, {14, 6, 10, 4, 6, 22}, {14, 6, 
            18, 4, 6, 16}, {14, 6, 18, 4, 8, 16}, {14, 6, 18, 4, 16, 8}, {14, 
            6, 20, 14, 10, 4}, {14, 8, 4, 14, 8, 20}, {14, 8, 4, 16, 10, 
            18}, {14, 8, 4, 18, 6, 16}, {14, 8, 4, 18, 8, 16}, {14, 8, 6, 6, 
            4, 22}, {14, 8, 6, 14, 4, 20}, {14, 8, 6, 18, 4, 16}, {14, 8, 10, 
            4, 18, 6}, {14, 8, 14, 4, 8, 20}, {14, 8, 14, 20, 8, 4}, {14, 8, 
            18, 4, 8, 16}, {14, 8, 18, 4, 16, 8}, {14, 8, 18, 12, 16, 
            4}, {14, 8, 18, 16, 12, 4}, {14, 8, 20, 14, 4, 6}, {14, 8, 20, 
            14, 8, 4}, {14, 10, 6, 6, 4, 22}, {14, 10, 8, 18, 4, 6}, {14, 10, 
            14, 4, 6, 20}, {14, 10, 14, 6, 4, 20}, {14, 10, 14, 20, 4, 
            6}, {14, 10, 14, 20, 6, 4}, {14, 10, 16, 4, 8, 18}, {14, 10, 16, 
            10, 18, 4}, {14, 10, 16, 18, 10, 4}, {14, 10, 18, 4, 6, 16}, {14, 
            10, 18, 4, 16, 6}, {14, 10, 18, 6, 4, 16}, {14, 10, 18, 10, 16, 
            4}, {14, 10, 18, 16, 10, 4}, {14, 12, 16, 4, 6, 18}, {14, 12, 16, 
            6, 4, 18}, {14, 12, 16, 8, 18, 4}, {14, 12, 16, 18, 8, 4}, {14, 
            14, 4, 8, 6, 20}, {14, 14, 4, 8, 20, 6}, {14, 14, 8, 8, 4, 
            20}, {14, 14, 8, 8, 20, 4}, {14, 14, 10, 4, 6, 20}, {14, 14, 10, 
            4, 20, 6}, {14, 14, 10, 6, 4, 20}, {14, 14, 10, 6, 20, 4}, {14, 
            16, 4, 18, 6, 8}, {14, 16, 4, 18, 8, 8}, {14, 16, 4, 18, 10, 
            6}, {14, 16, 10, 8, 4, 18}, {14, 16, 10, 10, 18, 4}, {14, 16, 10, 
            18, 10, 4}, {14, 16, 12, 4, 6, 18}, {14, 16, 12, 6, 4, 18}, {14, 
            16, 12, 8, 18, 4}, {14, 16, 12, 18, 8, 4}, {14, 18, 4, 8, 6, 
            16}, {14, 18, 4, 10, 8, 6}, {14, 18, 6, 6, 4, 16}, {14, 18, 6, 8, 
            4, 16}, {14, 18, 6, 16, 4, 8}, {14, 18, 8, 8, 4, 16}, {14, 18, 8, 
            12, 16, 4}, {14, 18, 8, 16, 4, 8}, {14, 18, 8, 16, 12, 4}, {14, 
            18, 10, 4, 6, 16}, {14, 18, 10, 6, 4, 16}, {14, 18, 10, 10, 16, 
            4}, {14, 18, 10, 16, 4, 6}, {14, 18, 10, 16, 10, 4}, {14, 20, 4, 
            10, 14, 6}, {14, 20, 6, 10, 14, 4}, {14, 20, 8, 4, 14, 6}, {14, 
            20, 8, 8, 14, 4}, {16, 4, 6, 6, 18, 14}, {16, 4, 6, 8, 18, 
            14}, {16, 4, 6, 18, 8, 14}, {16, 4, 8, 8, 18, 14}, {16, 4, 8, 10, 
            12, 20}, {16, 4, 8, 12, 14, 18}, {16, 4, 10, 6, 18, 14}, {16, 4, 
            10, 8, 12, 20}, {16, 4, 10, 10, 14, 18}, {16, 4, 12, 6, 18, 
            8}, {16, 4, 14, 6, 18, 10}, {16, 4, 14, 8, 18, 6}, {16, 4, 14, 8, 
            18, 8}, {16, 4, 18, 6, 10, 8}, {16, 4, 18, 6, 10, 14}, {16, 4, 
            18, 10, 14, 10}, {16, 4, 18, 12, 14, 8}, {16, 4, 18, 14, 12, 
            6}, {16, 6, 4, 8, 18, 14}, {16, 6, 4, 18, 6, 14}, {16, 6, 4, 18, 
            8, 14}, {16, 6, 10, 4, 18, 8}, {16, 6, 10, 4, 18, 14}, {16, 6, 
            18, 4, 6, 14}, {16, 6, 18, 4, 10, 14}, {16, 6, 18, 4, 12, 
            8}, {16, 6, 18, 4, 14, 10}, {16, 6, 18, 14, 12, 4}, {16, 8, 4, 
            12, 10, 20}, {16, 8, 4, 14, 12, 18}, {16, 8, 4, 18, 8, 14}, {16, 
            8, 12, 4, 10, 20}, {16, 8, 18, 4, 6, 14}, {16, 8, 18, 4, 8, 
            14}, {16, 8, 18, 4, 14, 6}, {16, 8, 18, 4, 14, 8}, {16, 8, 18, 6, 
            4, 14}, {16, 8, 18, 14, 10, 4}, {16, 10, 4, 12, 8, 20}, {16, 10, 
            4, 14, 10, 18}, {16, 10, 4, 18, 6, 14}, {16, 10, 6, 18, 4, 
            8}, {16, 10, 6, 18, 4, 14}, {16, 10, 12, 4, 8, 20}, {16, 10, 14, 
            4, 10, 18}, {16, 10, 14, 4, 18, 10}, {16, 10, 14, 18, 8, 4}, {16, 
            12, 4, 18, 6, 8}, {16, 12, 8, 10, 4, 20}, {16, 12, 10, 8, 4, 
            20}, {16, 12, 14, 4, 8, 18}, {16, 12, 14, 4, 18, 8}, {16, 12, 14, 
            18, 4, 6}, {16, 12, 14, 18, 6, 4}, {16, 14, 4, 18, 6, 10}, {16, 
            14, 4, 18, 8, 6}, {16, 14, 4, 18, 8, 8}, {16, 14, 10, 8, 18, 
            4}, {16, 14, 10, 10, 4, 18}, {16, 14, 10, 18, 4, 10}, {16, 14, 
            12, 4, 18, 6}, {16, 14, 12, 6, 18, 4}, {16, 14, 12, 8, 4, 
            18}, {16, 14, 12, 18, 4, 8}, {16, 18, 4, 10, 6, 8}, {16, 18, 4, 
            10, 6, 14}, {16, 18, 4, 12, 14, 6}, {16, 18, 4, 14, 10, 10}, {16, 
            18, 4, 14, 12, 8}, {16, 18, 6, 6, 4, 14}, {16, 18, 6, 10, 4, 
            14}, {16, 18, 6, 12, 4, 8}, {16, 18, 6, 12, 14, 4}, {16, 18, 6, 
            14, 4, 10}, {16, 18, 8, 4, 6, 14}, {16, 18, 8, 6, 4, 14}, {16, 
            18, 8, 8, 4, 14}, {16, 18, 8, 10, 14, 4}, {16, 18, 8, 14, 4, 
            6}, {16, 18, 8, 14, 4, 8}, {18, 4, 6, 12, 12, 18}, {18, 4, 10, 8, 
            16, 14}, {18, 4, 10, 10, 14, 16}, {18, 4, 12, 6, 12, 18}, {18, 4, 
            12, 6, 16, 14}, {18, 4, 12, 8, 14, 16}, {18, 4, 14, 6, 10, 
            8}, {18, 4, 14, 16, 8, 6}, {18, 4, 16, 6, 12, 14}, {18, 4, 16, 8, 
            10, 6}, {18, 4, 16, 8, 14, 12}, {18, 4, 16, 10, 14, 10}, {18, 4, 
            16, 14, 10, 6}, {18, 6, 4, 12, 12, 18}, {18, 6, 10, 4, 14, 
            8}, {18, 6, 12, 4, 12, 18}, {18, 6, 12, 4, 16, 14}, {18, 6, 14, 
            8, 16, 4}, {18, 6, 14, 16, 6, 4}, {18, 6, 14, 16, 8, 4}, {18, 6, 
            16, 4, 12, 14}, {18, 6, 16, 8, 12, 4}, {18, 6, 16, 10, 14, 
            4}, {18, 6, 16, 14, 6, 4}, {18, 6, 16, 14, 10, 4}, {18, 8, 10, 4, 
            16, 6}, {18, 8, 12, 6, 16, 4}, {18, 8, 14, 4, 12, 16}, {18, 8, 
            14, 4, 16, 12}, {18, 8, 14, 8, 16, 4}, {18, 8, 14, 16, 8, 
            4}, {18, 8, 16, 4, 10, 14}, {18, 8, 16, 6, 14, 4}, {18, 8, 16, 8, 
            14, 4}, {18, 8, 16, 14, 4, 6}, {18, 8, 16, 14, 6, 4}, {18, 8, 16, 
            14, 8, 4}, {18, 10, 4, 14, 10, 16}, {18, 10, 4, 16, 8, 14}, {18, 
            10, 6, 14, 4, 8}, {18, 10, 8, 16, 4, 6}, {18, 10, 14, 4, 10, 
            16}, {18, 10, 14, 4, 16, 10}, {18, 10, 14, 6, 16, 4}, {18, 10, 
            14, 16, 4, 6}, {18, 10, 14, 16, 6, 4}, {18, 12, 4, 12, 6, 
            18}, {18, 12, 4, 14, 8, 16}, {18, 12, 4, 16, 6, 14}, {18, 12, 6, 
            12, 4, 18}, {18, 12, 6, 16, 4, 14}, {18, 12, 8, 16, 6, 4}, {18, 
            12, 12, 4, 6, 18}, {18, 12, 12, 6, 4, 18}, {18, 14, 4, 8, 16, 
            6}, {18, 14, 4, 10, 6, 8}, {18, 14, 6, 6, 16, 4}, {18, 14, 6, 8, 
            16, 4}, {18, 14, 6, 16, 8, 4}, {18, 14, 8, 8, 16, 4}, {18, 14, 8, 
            12, 4, 16}, {18, 14, 8, 16, 4, 12}, {18, 14, 8, 16, 8, 4}, {18, 
            14, 10, 4, 16, 6}, {18, 14, 10, 6, 16, 4}, {18, 14, 10, 10, 4, 
            16}, {18, 14, 10, 16, 4, 10}, {18, 14, 10, 16, 6, 4}, {18, 16, 4, 
            10, 8, 6}, {18, 16, 4, 10, 14, 6}, {18, 16, 4, 12, 6, 14}, {18, 
            16, 4, 14, 8, 12}, {18, 16, 4, 14, 10, 10}, {18, 16, 6, 6, 14, 
            4}, {18, 16, 6, 10, 14, 4}, {18, 16, 6, 12, 4, 14}, {18, 16, 6, 
            12, 8, 4}, {18, 16, 6, 14, 10, 4}, {18, 16, 8, 4, 14, 6}, {18, 
            16, 8, 6, 14, 4}, {18, 16, 8, 8, 14, 4}, {18, 16, 8, 10, 4, 
            14}, {18, 16, 8, 14, 6, 4}, {18, 16, 8, 14, 8, 4}, {20, 4, 6, 14, 
            8, 14}, {20, 4, 8, 8, 14, 14}, {20, 4, 8, 10, 12, 16}, {20, 4, 
            10, 6, 14, 14}, {20, 4, 10, 8, 12, 16}, {20, 4, 14, 6, 10, 
            14}, {20, 6, 4, 8, 14, 14}, {20, 6, 10, 4, 14, 14}, {20, 6, 14, 
            4, 10, 14}, {20, 8, 4, 12, 10, 16}, {20, 8, 4, 14, 8, 14}, {20, 
            8, 12, 4, 10, 16}, {20, 8, 14, 4, 8, 14}, {20, 8, 14, 6, 4, 
            14}, {20, 10, 4, 12, 8, 16}, {20, 10, 4, 14, 6, 14}, {20, 10, 6, 
            14, 4, 14}, {20, 10, 12, 4, 8, 16}, {20, 12, 8, 10, 4, 16}, {20, 
            12, 10, 8, 4, 16}, {20, 14, 4, 10, 6, 14}, {20, 14, 6, 10, 4, 
            14}, {20, 14, 8, 4, 6, 14}, {20, 14, 8, 8, 4, 14}, {22, 4, 6, 6, 
            8, 14}, {22, 4, 6, 6, 10, 14}, {22, 4, 8, 6, 12, 12}, {22, 4, 8, 
            8, 12, 12}, {22, 4, 8, 12, 8, 12}, {22, 4, 10, 6, 6, 12}, {22, 4, 
            10, 6, 12, 12}, {22, 4, 10, 8, 12, 10}, {22, 4, 10, 12, 8, 
            10}, {22, 4, 12, 6, 6, 12}, {22, 4, 12, 8, 6, 10}, {22, 4, 12, 8, 
            10, 12}, {22, 4, 12, 10, 8, 8}, {22, 4, 12, 10, 10, 8}, {22, 4, 
            12, 10, 10, 10}, {22, 6, 4, 8, 6, 14}, {22, 6, 4, 10, 6, 
            14}, {22, 6, 6, 4, 10, 12}, {22, 6, 6, 4, 12, 12}, {22, 6, 6, 10, 
            4, 12}, {22, 6, 6, 12, 4, 12}, {22, 6, 8, 4, 6, 14}, {22, 6, 8, 
            12, 4, 10}, {22, 6, 10, 4, 6, 14}, {22, 6, 12, 4, 8, 12}, {22, 6, 
            12, 4, 10, 12}, {22, 8, 4, 8, 12, 12}, {22, 8, 4, 12, 6, 
            12}, {22, 8, 4, 12, 8, 12}, {22, 8, 6, 4, 12, 10}, {22, 8, 6, 6, 
            4, 14}, {22, 8, 10, 4, 12, 12}, {22, 8, 10, 12, 4, 8}, {22, 8, 
            12, 4, 8, 12}, {22, 8, 12, 4, 10, 10}, {22, 8, 12, 8, 4, 
            12}, {22, 8, 12, 10, 4, 10}, {22, 10, 4, 6, 6, 12}, {22, 10, 4, 
            8, 12, 10}, {22, 10, 4, 12, 6, 12}, {22, 10, 4, 12, 8, 10}, {22, 
            10, 6, 6, 4, 14}, {22, 10, 8, 4, 12, 8}, {22, 10, 8, 12, 4, 
            12}, {22, 10, 10, 4, 12, 8}, {22, 10, 10, 4, 12, 10}, {22, 10, 
            10, 12, 4, 8}, {22, 10, 10, 12, 4, 10}, {22, 12, 4, 6, 6, 
            12}, {22, 12, 4, 6, 8, 10}, {22, 12, 4, 8, 10, 8}, {22, 12, 4, 
            10, 8, 12}, {22, 12, 4, 10, 10, 8}, {22, 12, 4, 10, 10, 10}, {22, 
            12, 6, 8, 4, 12}, {22, 12, 6, 10, 4, 12}, {22, 12, 8, 4, 8, 
            12}, {22, 12, 8, 4, 10, 10}, {22, 12, 8, 8, 4, 12}, {22, 12, 8, 
            10, 4, 10}, {4, 4, 10, 16, 12, 18}, {4, 4, 10, 16, 14, 18}, {4, 
            4, 10, 16, 18, 12}, {4, 4, 10, 16, 18, 14}, {4, 4, 10, 18, 10, 
            14}, {4, 4, 10, 18, 10, 16}, {4, 4, 10, 18, 14, 10}, {4, 4, 10, 
            18, 16, 10}, {4, 4, 12, 20, 8, 12}, {4, 4, 12, 20, 8, 14}, {4, 4, 
            12, 20, 12, 8}, {4, 4, 12, 20, 14, 8}, {4, 4, 14, 16, 8, 18}, {4, 
            4, 14, 16, 10, 18}, {4, 4, 14, 16, 18, 8}, {4, 4, 14, 16, 18, 
            10}, {4, 4, 14, 18, 6, 14}, {4, 4, 14, 18, 6, 16}, {4, 4, 14, 18, 
            14, 6}, {4, 4, 14, 18, 16, 6}, {4, 4, 14, 20, 6, 12}, {4, 4, 14, 
            20, 6, 14}, {4, 4, 14, 20, 12, 6}, {4, 4, 14, 20, 14, 6}, {4, 6, 
            14, 16, 16, 4}, {4, 6, 14, 16, 18, 4}, {4, 6, 14, 20, 12, 4}, {4, 
            6, 14, 20, 14, 4}, {4, 6, 16, 14, 16, 4}, {4, 6, 16, 14, 18, 
            4}, {4, 6, 18, 14, 4, 14}, {4, 6, 18, 14, 4, 16}, {4, 6, 20, 14, 
            4, 12}, {4, 6, 20, 14, 4, 14}, {4, 6, 20, 14, 12, 4}, {4, 6, 20, 
            14, 14, 4}, {4, 8, 12, 22, 10, 4}, {4, 8, 12, 22, 12, 4}, {4, 8, 
            16, 14, 4, 18}, {4, 8, 20, 12, 4, 12}, {4, 8, 20, 12, 4, 14}, {4, 
            8, 22, 12, 10, 4}, {4, 8, 22, 12, 12, 4}, {4, 10, 4, 10, 18, 
            14}, {4, 10, 4, 10, 18, 16}, {4, 10, 4, 12, 16, 18}, {4, 10, 4, 
            14, 16, 18}, {4, 10, 4, 14, 18, 10}, {4, 10, 4, 16, 18, 10}, {4, 
            10, 4, 18, 16, 12}, {4, 10, 4, 18, 16, 14}, {4, 10, 10, 12, 22, 
            4}, {4, 10, 10, 22, 12, 4}, {4, 10, 12, 10, 22, 4}, {4, 10, 12, 
            22, 8, 4}, {4, 10, 16, 14, 4, 18}, {4, 10, 18, 10, 4, 14}, {4, 
            10, 18, 10, 4, 16}, {4, 10, 22, 10, 12, 4}, {4, 10, 22, 12, 8, 
            4}, {4, 12, 4, 8, 20, 12}, {4, 12, 4, 8, 20, 14}, {4, 12, 4, 12, 
            20, 8}, {4, 12, 4, 14, 20, 8}, {4, 12, 8, 10, 22, 4}, {4, 12, 8, 
            12, 22, 4}, {4, 12, 10, 8, 22, 4}, {4, 12, 10, 22, 10, 4}, {4, 
            12, 12, 8, 22, 4}, {4, 12, 12, 22, 8, 4}, {4, 12, 14, 20, 6, 
            4}, {4, 12, 16, 10, 4, 18}, {4, 12, 20, 12, 4, 8}, {4, 12, 20, 
            14, 4, 6}, {4, 12, 20, 14, 6, 4}, {4, 12, 22, 10, 10, 4}, {4, 12, 
            22, 12, 8, 4}, {4, 14, 4, 6, 18, 14}, {4, 14, 4, 6, 18, 16}, {4, 
            14, 4, 6, 20, 12}, {4, 14, 4, 6, 20, 14}, {4, 14, 4, 8, 16, 
            18}, {4, 14, 4, 10, 16, 18}, {4, 14, 4, 12, 20, 6}, {4, 14, 4, 
            14, 18, 6}, {4, 14, 4, 14, 20, 6}, {4, 14, 4, 16, 18, 6}, {4, 14, 
            4, 18, 16, 8}, {4, 14, 4, 18, 16, 10}, {4, 14, 6, 12, 20, 4}, {4, 
            14, 6, 14, 20, 4}, {4, 14, 6, 16, 16, 4}, {4, 14, 6, 18, 16, 
            4}, {4, 14, 12, 6, 20, 4}, {4, 14, 14, 6, 20, 4}, {4, 14, 14, 20, 
            6, 4}, {4, 14, 16, 6, 16, 4}, {4, 14, 16, 10, 4, 18}, {4, 14, 18, 
            6, 16, 4}, {4, 14, 18, 10, 4, 10}, {4, 14, 18, 14, 4, 6}, {4, 14, 
            20, 12, 4, 8}, {4, 14, 20, 14, 4, 6}, {4, 14, 20, 14, 6, 4}, {4, 
            16, 6, 16, 14, 4}, {4, 16, 6, 18, 14, 4}, {4, 16, 8, 4, 14, 
            18}, {4, 16, 10, 4, 14, 18}, {4, 16, 12, 4, 10, 18}, {4, 16, 14, 
            4, 10, 18}, {4, 16, 14, 16, 6, 4}, {4, 16, 16, 6, 14, 4}, {4, 16, 
            16, 14, 6, 4}, {4, 16, 18, 4, 10, 12}, {4, 16, 18, 4, 10, 
            14}, {4, 16, 18, 4, 14, 8}, {4, 16, 18, 4, 14, 10}, {4, 16, 18, 
            6, 14, 4}, {4, 16, 18, 10, 4, 10}, {4, 16, 18, 14, 4, 6}, {4, 18, 
            6, 4, 14, 14}, {4, 18, 6, 4, 14, 16}, {4, 18, 10, 4, 10, 14}, {4, 
            18, 10, 4, 10, 16}, {4, 18, 14, 4, 10, 10}, {4, 18, 14, 4, 14, 
            6}, {4, 18, 14, 16, 6, 4}, {4, 18, 16, 4, 10, 10}, {4, 18, 16, 4, 
            14, 6}, {4, 18, 16, 10, 4, 12}, {4, 18, 16, 10, 4, 14}, {4, 18, 
            16, 14, 4, 8}, {4, 18, 16, 14, 4, 10}, {4, 18, 16, 14, 6, 4}, {4, 
            20, 6, 4, 14, 12}, {4, 20, 6, 4, 14, 14}, {4, 20, 6, 12, 14, 
            4}, {4, 20, 6, 14, 14, 4}, {4, 20, 8, 4, 12, 12}, {4, 20, 8, 4, 
            12, 14}, {4, 20, 12, 4, 12, 8}, {4, 20, 12, 4, 14, 6}, {4, 20, 
            12, 6, 14, 4}, {4, 20, 14, 4, 12, 8}, {4, 20, 14, 4, 14, 6}, {4, 
            20, 14, 6, 14, 4}, {4, 22, 8, 10, 12, 4}, {4, 22, 8, 12, 12, 
            4}, {4, 22, 10, 8, 12, 4}, {4, 22, 10, 12, 10, 4}, {4, 22, 12, 8, 
            12, 4}, {4, 22, 12, 10, 10, 4}, {6, 4, 14, 16, 4, 16}, {6, 4, 14, 
            16, 4, 18}, {6, 4, 14, 20, 4, 12}, {6, 4, 14, 20, 4, 14}, {6, 4, 
            16, 14, 4, 16}, {6, 4, 16, 14, 4, 18}, {6, 4, 18, 14, 14, 4}, {6, 
            4, 18, 14, 16, 4}, {6, 4, 20, 14, 4, 12}, {6, 4, 20, 14, 4, 
            14}, {6, 4, 20, 14, 12, 4}, {6, 4, 20, 14, 14, 4}, {6, 12, 14, 
            20, 4, 4}, {6, 14, 4, 4, 16, 16}, {6, 14, 4, 4, 16, 18}, {6, 14, 
            4, 4, 20, 12}, {6, 14, 4, 4, 20, 14}, {6, 14, 12, 4, 20, 4}, {6, 
            14, 14, 4, 18, 4}, {6, 14, 14, 4, 20, 4}, {6, 14, 14, 18, 4, 
            4}, {6, 14, 14, 20, 4, 4}, {6, 14, 16, 4, 18, 4}, {6, 16, 4, 4, 
            14, 16}, {6, 16, 4, 4, 14, 18}, {6, 16, 14, 18, 4, 4}, {6, 18, 4, 
            14, 14, 4}, {6, 18, 4, 16, 14, 4}, {6, 20, 4, 4, 14, 12}, {6, 20, 
            4, 4, 14, 14}, {6, 20, 4, 12, 14, 4}, {6, 20, 4, 14, 14, 4}, {8, 
            4, 12, 22, 4, 10}, {8, 4, 12, 22, 4, 12}, {8, 4, 16, 14, 18, 
            4}, {8, 4, 20, 12, 12, 4}, {8, 4, 20, 12, 14, 4}, {8, 4, 22, 12, 
            4, 10}, {8, 4, 22, 12, 4, 12}, {8, 12, 4, 4, 22, 10}, {8, 12, 4, 
            4, 22, 12}, {8, 12, 12, 4, 20, 4}, {8, 12, 12, 20, 4, 4}, {8, 12, 
            14, 4, 20, 4}, {8, 14, 12, 20, 4, 4}, {8, 14, 18, 4, 16, 4}, {8, 
            16, 4, 18, 14, 4}, {8, 18, 14, 16, 4, 4}, {8, 20, 4, 12, 12, 
            4}, {8, 20, 4, 14, 12, 4}, {8, 22, 4, 4, 12, 10}, {8, 22, 4, 4, 
            12, 12}, {10, 4, 4, 10, 14, 18}, {10, 4, 4, 10, 16, 18}, {10, 4, 
            4, 12, 18, 16}, {10, 4, 4, 14, 10, 18}, {10, 4, 4, 14, 18, 
            16}, {10, 4, 4, 16, 10, 18}, {10, 4, 4, 18, 12, 16}, {10, 4, 4, 
            18, 14, 16}, {10, 4, 10, 12, 4, 22}, {10, 4, 10, 22, 4, 12}, {10, 
            4, 12, 10, 4, 22}, {10, 4, 12, 22, 4, 8}, {10, 4, 16, 14, 18, 
            4}, {10, 4, 18, 10, 14, 4}, {10, 4, 18, 10, 16, 4}, {10, 4, 22, 
            10, 4, 12}, {10, 4, 22, 12, 4, 8}, {10, 10, 4, 4, 12, 22}, {10, 
            10, 4, 4, 22, 12}, {10, 10, 14, 4, 4, 18}, {10, 10, 14, 4, 18, 
            4}, {10, 10, 16, 4, 4, 18}, {10, 10, 16, 4, 18, 4}, {10, 12, 4, 
            4, 10, 22}, {10, 12, 4, 4, 22, 8}, {10, 12, 18, 4, 4, 16}, {10, 
            14, 10, 4, 4, 18}, {10, 14, 10, 18, 4, 4}, {10, 14, 18, 4, 4, 
            16}, {10, 14, 18, 4, 16, 4}, {10, 16, 4, 18, 14, 4}, {10, 16, 10, 
            4, 4, 18}, {10, 16, 10, 18, 4, 4}, {10, 18, 4, 14, 10, 4}, {10, 
            18, 4, 16, 10, 4}, {10, 18, 12, 4, 4, 16}, {10, 18, 14, 4, 4, 
            16}, {10, 18, 14, 16, 4, 4}, {10, 22, 4, 4, 10, 12}, {10, 22, 4, 
            4, 12, 8}, {12, 4, 4, 8, 12, 20}, {12, 4, 4, 8, 14, 20}, {12, 4, 
            4, 12, 8, 20}, {12, 4, 4, 14, 8, 20}, {12, 4, 8, 10, 4, 22}, {12, 
            4, 8, 12, 4, 22}, {12, 4, 10, 8, 4, 22}, {12, 4, 10, 22, 4, 
            10}, {12, 4, 12, 8, 4, 22}, {12, 4, 12, 22, 4, 8}, {12, 4, 14, 
            20, 4, 6}, {12, 4, 16, 10, 18, 4}, {12, 4, 20, 12, 8, 4}, {12, 4, 
            20, 14, 4, 6}, {12, 4, 20, 14, 6, 4}, {12, 4, 22, 10, 4, 
            10}, {12, 4, 22, 12, 4, 8}, {12, 6, 14, 20, 4, 4}, {12, 8, 4, 4, 
            10, 22}, {12, 8, 4, 4, 12, 22}, {12, 8, 12, 4, 4, 20}, {12, 8, 
            12, 20, 4, 4}, {12, 8, 14, 4, 4, 20}, {12, 10, 4, 4, 8, 22}, {12, 
            10, 4, 4, 22, 10}, {12, 10, 18, 4, 16, 4}, {12, 12, 4, 4, 8, 
            22}, {12, 12, 4, 4, 22, 8}, {12, 12, 8, 4, 4, 20}, {12, 12, 8, 4, 
            20, 4}, {12, 14, 4, 4, 20, 6}, {12, 14, 6, 4, 20, 4}, {12, 14, 8, 
            4, 4, 20}, {12, 16, 4, 18, 10, 4}, {12, 18, 10, 16, 4, 4}, {12, 
            20, 4, 4, 14, 6}, {12, 20, 4, 6, 14, 4}, {12, 20, 4, 8, 12, 
            4}, {12, 22, 4, 4, 10, 10}, {12, 22, 4, 4, 12, 8}, {14, 4, 4, 6, 
            12, 20}, {14, 4, 4, 6, 14, 18}, {14, 4, 4, 6, 14, 20}, {14, 4, 4, 
            6, 16, 18}, {14, 4, 4, 8, 18, 16}, {14, 4, 4, 10, 18, 16}, {14, 
            4, 4, 12, 6, 20}, {14, 4, 4, 14, 6, 18}, {14, 4, 4, 14, 6, 
            20}, {14, 4, 4, 16, 6, 18}, {14, 4, 4, 18, 8, 16}, {14, 4, 4, 18, 
            10, 16}, {14, 4, 6, 12, 4, 20}, {14, 4, 6, 14, 4, 20}, {14, 4, 6, 
            16, 4, 16}, {14, 4, 6, 18, 4, 16}, {14, 4, 12, 6, 4, 20}, {14, 4, 
            14, 6, 4, 20}, {14, 4, 14, 20, 4, 6}, {14, 4, 16, 6, 4, 16}, {14, 
            4, 16, 10, 18, 4}, {14, 4, 18, 6, 4, 16}, {14, 4, 18, 10, 10, 
            4}, {14, 4, 18, 14, 6, 4}, {14, 4, 20, 12, 8, 4}, {14, 4, 20, 14, 
            4, 6}, {14, 4, 20, 14, 6, 4}, {14, 6, 4, 4, 12, 20}, {14, 6, 4, 
            4, 14, 20}, {14, 6, 4, 4, 16, 16}, {14, 6, 4, 4, 18, 16}, {14, 6, 
            12, 4, 4, 20}, {14, 6, 14, 4, 4, 18}, {14, 6, 14, 4, 4, 20}, {14, 
            6, 14, 18, 4, 4}, {14, 6, 14, 20, 4, 4}, {14, 6, 16, 4, 4, 
            18}, {14, 8, 12, 20, 4, 4}, {14, 8, 18, 4, 4, 16}, {14, 10, 10, 
            4, 18, 4}, {14, 10, 10, 18, 4, 4}, {14, 10, 18, 4, 4, 16}, {14, 
            10, 18, 4, 16, 4}, {14, 12, 4, 4, 6, 20}, {14, 12, 6, 4, 4, 
            20}, {14, 12, 8, 4, 20, 4}, {14, 14, 4, 4, 6, 20}, {14, 14, 4, 4, 
            20, 6}, {14, 14, 6, 4, 4, 18}, {14, 14, 6, 4, 4, 20}, {14, 14, 6, 
            4, 18, 4}, {14, 14, 6, 4, 20, 4}, {14, 16, 4, 4, 6, 16}, {14, 16, 
            4, 18, 10, 4}, {14, 16, 6, 4, 4, 18}, {14, 18, 4, 4, 6, 16}, {14, 
            18, 4, 6, 14, 4}, {14, 18, 4, 10, 10, 4}, {14, 18, 8, 4, 4, 
            16}, {14, 18, 10, 4, 4, 16}, {14, 18, 10, 16, 4, 4}, {14, 20, 4, 
            4, 14, 6}, {14, 20, 4, 6, 14, 4}, {14, 20, 4, 8, 12, 4}, {16, 4, 
            6, 16, 4, 14}, {16, 4, 6, 18, 4, 14}, {16, 4, 8, 4, 18, 14}, {16, 
            4, 10, 4, 18, 14}, {16, 4, 12, 4, 18, 10}, {16, 4, 14, 4, 18, 
            10}, {16, 4, 14, 16, 4, 6}, {16, 4, 16, 6, 4, 14}, {16, 4, 16, 
            14, 4, 6}, {16, 4, 18, 4, 8, 14}, {16, 4, 18, 4, 10, 14}, {16, 4, 
            18, 4, 12, 10}, {16, 4, 18, 4, 14, 10}, {16, 4, 18, 6, 4, 
            14}, {16, 4, 18, 10, 10, 4}, {16, 4, 18, 14, 6, 4}, {16, 6, 4, 4, 
            16, 14}, {16, 6, 4, 4, 18, 14}, {16, 6, 14, 18, 4, 4}, {16, 8, 4, 
            18, 4, 14}, {16, 10, 4, 18, 4, 14}, {16, 10, 10, 4, 18, 4}, {16, 
            10, 10, 18, 4, 4}, {16, 12, 4, 18, 4, 10}, {16, 14, 4, 4, 16, 
            6}, {16, 14, 4, 18, 4, 10}, {16, 14, 6, 4, 18, 4}, {16, 16, 4, 4, 
            6, 14}, {16, 16, 4, 4, 14, 6}, {16, 18, 4, 4, 6, 14}, {16, 18, 4, 
            6, 14, 4}, {16, 18, 4, 8, 4, 14}, {16, 18, 4, 10, 4, 14}, {16, 
            18, 4, 10, 10, 4}, {16, 18, 4, 12, 4, 10}, {16, 18, 4, 14, 4, 
            10}, {18, 4, 6, 4, 14, 14}, {18, 4, 6, 4, 16, 14}, {18, 4, 10, 4, 
            14, 10}, {18, 4, 10, 4, 16, 10}, {18, 4, 14, 4, 6, 14}, {18, 4, 
            14, 4, 10, 10}, {18, 4, 14, 16, 4, 6}, {18, 4, 16, 4, 6, 
            14}, {18, 4, 16, 4, 10, 10}, {18, 4, 16, 10, 12, 4}, {18, 4, 16, 
            10, 14, 4}, {18, 4, 16, 14, 4, 6}, {18, 4, 16, 14, 8, 4}, {18, 4, 
            16, 14, 10, 4}, {18, 6, 4, 14, 4, 14}, {18, 6, 4, 16, 4, 
            14}, {18, 8, 14, 16, 4, 4}, {18, 10, 4, 14, 4, 10}, {18, 10, 4, 
            16, 4, 10}, {18, 10, 12, 4, 16, 4}, {18, 10, 14, 4, 16, 4}, {18, 
            10, 14, 16, 4, 4}, {18, 12, 10, 16, 4, 4}, {18, 14, 4, 4, 16, 
            6}, {18, 14, 4, 6, 4, 14}, {18, 14, 4, 10, 4, 10}, {18, 14, 8, 4, 
            16, 4}, {18, 14, 10, 4, 16, 4}, {18, 14, 10, 16, 4, 4}, {18, 16, 
            4, 4, 14, 6}, {18, 16, 4, 6, 4, 14}, {18, 16, 4, 8, 14, 4}, {18, 
            16, 4, 10, 4, 10}, {18, 16, 4, 10, 14, 4}, {18, 16, 4, 12, 10, 
            4}, {18, 16, 4, 14, 10, 4}, {20, 4, 6, 4, 12, 14}, {20, 4, 6, 4, 
            14, 14}, {20, 4, 6, 12, 4, 14}, {20, 4, 6, 14, 4, 14}, {20, 4, 8, 
            4, 12, 12}, {20, 4, 8, 4, 14, 12}, {20, 4, 12, 4, 6, 14}, {20, 4, 
            12, 4, 8, 12}, {20, 4, 12, 6, 4, 14}, {20, 4, 14, 4, 6, 14}, {20, 
            4, 14, 4, 8, 12}, {20, 4, 14, 6, 4, 14}, {20, 6, 4, 4, 12, 
            14}, {20, 6, 4, 4, 14, 14}, {20, 6, 4, 12, 4, 14}, {20, 6, 4, 14, 
            4, 14}, {20, 8, 4, 12, 4, 12}, {20, 8, 4, 14, 4, 12}, {20, 12, 4, 
            4, 6, 14}, {20, 12, 4, 6, 4, 14}, {20, 12, 4, 8, 4, 12}, {20, 14, 
            4, 4, 6, 14}, {20, 14, 4, 6, 4, 14}, {20, 14, 4, 8, 4, 12}, {22, 
            4, 8, 10, 4, 12}, {22, 4, 8, 12, 4, 12}, {22, 4, 10, 8, 4, 
            12}, {22, 4, 10, 12, 4, 10}, {22, 4, 12, 8, 4, 12}, {22, 4, 12, 
            10, 4, 10}, {22, 8, 4, 4, 10, 12}, {22, 8, 4, 4, 12, 12}, {22, 
            10, 4, 4, 8, 12}, {22, 10, 4, 4, 12, 10}, {22, 12, 4, 4, 8, 
            12}, {22, 12, 4, 4, 10, 10}, {2, 6, 10, 16, 18, 10}, {2, 6, 10, 
            18, 12, 16}, {2, 6, 12, 18, 14, 16}, {2, 6, 12, 20, 10, 14}, {2, 
            6, 12, 20, 12, 14}, {2, 6, 14, 16, 16, 6}, {2, 6, 14, 16, 18, 
            6}, {2, 6, 14, 16, 18, 10}, {2, 6, 14, 18, 8, 16}, {2, 6, 14, 18, 
            12, 16}, {2, 6, 14, 20, 8, 14}, {2, 6, 14, 20, 10, 14}, {2, 6, 
            14, 20, 12, 6}, {2, 6, 14, 20, 14, 6}, {2, 6, 14, 20, 14, 
            10}, {2, 6, 16, 18, 6, 12}, {2, 6, 16, 18, 6, 14}, {2, 6, 16, 20, 
            6, 10}, {2, 6, 16, 20, 6, 12}, {2, 6, 16, 20, 10, 6}, {2, 6, 16, 
            20, 12, 6}, {2, 6, 16, 20, 12, 10}, {2, 6, 18, 16, 6, 12}, {2, 6, 
            18, 16, 6, 14}, {2, 6, 20, 16, 6, 10}, {2, 6, 20, 16, 6, 12}, {2, 
            6, 20, 16, 10, 6}, {2, 6, 20, 16, 12, 6}, {2, 6, 20, 16, 12, 
            10}, {2, 8, 8, 14, 18, 10}, {2, 8, 8, 14, 18, 16}, {2, 8, 8, 18, 
            14, 10}, {2, 8, 8, 18, 14, 16}, {2, 8, 10, 18, 14, 16}, {2, 8, 
            10, 20, 12, 14}, {2, 8, 12, 22, 10, 6}, {2, 8, 12, 22, 10, 
            10}, {2, 8, 12, 22, 12, 6}, {2, 8, 12, 22, 12, 10}, {2, 8, 14, 
            18, 8, 16}, {2, 8, 14, 18, 10, 16}, {2, 8, 14, 20, 8, 14}, {2, 8, 
            14, 22, 8, 6}, {2, 8, 14, 22, 8, 10}, {2, 8, 14, 22, 10, 6}, {2, 
            8, 14, 22, 10, 10}, {2, 8, 18, 14, 6, 16}, {2, 8, 18, 14, 8, 
            16}, {2, 8, 20, 14, 6, 14}, {2, 8, 20, 14, 8, 14}, {2, 8, 22, 14, 
            8, 6}, {2, 8, 22, 14, 8, 10}, {2, 8, 22, 14, 10, 6}, {2, 8, 22, 
            14, 10, 10}, {2, 10, 6, 12, 18, 16}, {2, 10, 6, 18, 16, 10}, {2, 
            10, 8, 12, 20, 14}, {2, 10, 8, 14, 18, 16}, {2, 10, 10, 12, 22, 
            6}, {2, 10, 10, 12, 22, 10}, {2, 10, 10, 22, 12, 6}, {2, 10, 10, 
            22, 12, 10}, {2, 10, 14, 22, 8, 6}, {2, 10, 14, 22, 8, 10}, {2, 
            10, 16, 20, 6, 6}, {2, 10, 18, 14, 8, 16}, {2, 10, 20, 12, 6, 
            14}, {2, 10, 20, 14, 6, 14}, {2, 10, 20, 16, 6, 6}, {2, 10, 22, 
            12, 8, 6}, {2, 10, 22, 12, 8, 10}, {2, 10, 22, 14, 8, 6}, {2, 10, 
            22, 14, 8, 10}, {2, 12, 6, 10, 20, 14}, {2, 12, 6, 12, 20, 
            14}, {2, 12, 6, 14, 18, 16}, {2, 12, 8, 10, 22, 6}, {2, 12, 8, 
            10, 22, 10}, {2, 12, 8, 12, 22, 6}, {2, 12, 8, 12, 22, 10}, {2, 
            12, 16, 20, 6, 6}, {2, 12, 16, 20, 6, 10}, {2, 12, 18, 10, 6, 
            16}, {2, 12, 18, 14, 6, 16}, {2, 12, 20, 10, 8, 14}, {2, 12, 20, 
            12, 6, 14}, {2, 12, 20, 14, 6, 6}, {2, 12, 20, 16, 6, 6}, {2, 12, 
            20, 16, 6, 10}, {2, 12, 22, 10, 10, 6}, {2, 12, 22, 10, 10, 
            10}, {2, 12, 22, 12, 8, 6}, {2, 12, 22, 12, 8, 10}, {2, 14, 6, 8, 
            18, 16}, {2, 14, 6, 8, 20, 14}, {2, 14, 6, 10, 20, 14}, {2, 14, 
            6, 12, 18, 16}, {2, 14, 6, 12, 20, 6}, {2, 14, 6, 14, 20, 6}, {2, 
            14, 6, 14, 20, 10}, {2, 14, 6, 16, 16, 6}, {2, 14, 6, 18, 16, 
            6}, {2, 14, 6, 18, 16, 10}, {2, 14, 8, 8, 18, 16}, {2, 14, 8, 8, 
            20, 14}, {2, 14, 8, 8, 22, 6}, {2, 14, 8, 8, 22, 10}, {2, 14, 8, 
            10, 18, 16}, {2, 14, 8, 10, 22, 6}, {2, 14, 8, 10, 22, 10}, {2, 
            14, 10, 8, 22, 6}, {2, 14, 10, 8, 22, 10}, {2, 14, 18, 8, 8, 
            10}, {2, 14, 18, 8, 8, 16}, {2, 14, 18, 10, 8, 16}, {2, 14, 18, 
            12, 6, 16}, {2, 14, 20, 14, 6, 6}, {2, 14, 20, 14, 6, 10}, {2, 
            16, 6, 6, 18, 12}, {2, 16, 6, 6, 18, 14}, {2, 16, 6, 6, 20, 
            10}, {2, 16, 6, 6, 20, 12}, {2, 16, 6, 10, 20, 6}, {2, 16, 6, 12, 
            20, 6}, {2, 16, 6, 12, 20, 10}, {2, 16, 10, 6, 20, 6}, {2, 16, 
            12, 6, 20, 6}, {2, 16, 12, 6, 20, 10}, {2, 16, 16, 6, 14, 6}, {2, 
            16, 16, 14, 6, 6}, {2, 16, 18, 6, 10, 10}, {2, 16, 18, 6, 14, 
            6}, {2, 16, 18, 6, 14, 10}, {2, 18, 6, 6, 16, 12}, {2, 18, 6, 6, 
            16, 14}, {2, 18, 8, 6, 14, 16}, {2, 18, 8, 8, 14, 16}, {2, 18, 
            10, 8, 14, 16}, {2, 18, 12, 6, 10, 16}, {2, 18, 12, 6, 14, 
            16}, {2, 18, 14, 6, 12, 16}, {2, 18, 14, 8, 8, 10}, {2, 18, 14, 
            8, 8, 16}, {2, 18, 14, 8, 10, 16}, {2, 18, 16, 10, 6, 10}, {2, 
            18, 16, 14, 6, 6}, {2, 18, 16, 14, 6, 10}, {2, 20, 6, 6, 16, 
            10}, {2, 20, 6, 6, 16, 12}, {2, 20, 6, 10, 16, 6}, {2, 20, 6, 12, 
            16, 6}, {2, 20, 6, 12, 16, 10}, {2, 20, 8, 6, 14, 14}, {2, 20, 8, 
            8, 14, 14}, {2, 20, 10, 6, 12, 14}, {2, 20, 10, 6, 14, 14}, {2, 
            20, 10, 6, 16, 6}, {2, 20, 12, 6, 12, 14}, {2, 20, 12, 6, 14, 
            6}, {2, 20, 12, 6, 16, 6}, {2, 20, 12, 6, 16, 10}, {2, 20, 12, 8, 
            10, 14}, {2, 20, 14, 6, 14, 6}, {2, 20, 14, 6, 14, 10}, {2, 22, 
            8, 8, 14, 6}, {2, 22, 8, 8, 14, 10}, {2, 22, 8, 10, 14, 6}, {2, 
            22, 8, 10, 14, 10}, {2, 22, 10, 8, 12, 6}, {2, 22, 10, 8, 12, 
            10}, {2, 22, 10, 8, 14, 6}, {2, 22, 10, 8, 14, 10}, {2, 22, 12, 
            8, 12, 6}, {2, 22, 12, 8, 12, 10}, {2, 22, 12, 10, 10, 6}, {2, 
            22, 12, 10, 10, 10}, {6, 2, 10, 16, 10, 18}, {6, 2, 10, 18, 16, 
            12}, {6, 2, 12, 18, 16, 14}, {6, 2, 12, 20, 14, 10}, {6, 2, 12, 
            20, 14, 12}, {6, 2, 14, 16, 6, 16}, {6, 2, 14, 16, 6, 18}, {6, 2, 
            14, 16, 10, 18}, {6, 2, 14, 18, 16, 8}, {6, 2, 14, 18, 16, 
            12}, {6, 2, 14, 20, 6, 12}, {6, 2, 14, 20, 6, 14}, {6, 2, 14, 20, 
            10, 14}, {6, 2, 14, 20, 14, 8}, {6, 2, 14, 20, 14, 10}, {6, 2, 
            16, 18, 12, 6}, {6, 2, 16, 18, 14, 6}, {6, 2, 16, 20, 6, 10}, {6, 
            2, 16, 20, 6, 12}, {6, 2, 16, 20, 10, 6}, {6, 2, 16, 20, 10, 
            12}, {6, 2, 16, 20, 12, 6}, {6, 2, 18, 16, 12, 6}, {6, 2, 18, 16, 
            14, 6}, {6, 2, 20, 16, 6, 10}, {6, 2, 20, 16, 6, 12}, {6, 2, 20, 
            16, 10, 6}, {6, 2, 20, 16, 10, 12}, {6, 2, 20, 16, 12, 6}, {6, 6, 
            16, 14, 2, 16}, {6, 6, 16, 14, 2, 18}, {6, 6, 16, 14, 16, 2}, {6, 
            6, 16, 14, 18, 2}, {6, 6, 16, 20, 2, 10}, {6, 6, 16, 20, 2, 
            12}, {6, 6, 16, 20, 10, 2}, {6, 6, 16, 20, 12, 2}, {6, 6, 20, 14, 
            2, 12}, {6, 6, 20, 14, 2, 14}, {6, 6, 20, 14, 12, 2}, {6, 6, 20, 
            14, 14, 2}, {6, 6, 20, 16, 2, 10}, {6, 6, 20, 16, 2, 12}, {6, 6, 
            20, 16, 10, 2}, {6, 6, 20, 16, 12, 2}, {6, 8, 14, 22, 8, 2}, {6, 
            8, 14, 22, 10, 2}, {6, 8, 22, 12, 10, 2}, {6, 8, 22, 12, 12, 
            2}, {6, 8, 22, 14, 8, 2}, {6, 8, 22, 14, 10, 2}, {6, 10, 2, 10, 
            16, 18}, {6, 10, 2, 16, 18, 12}, {6, 10, 12, 10, 22, 2}, {6, 10, 
            12, 22, 8, 2}, {6, 10, 14, 22, 8, 2}, {6, 10, 16, 10, 2, 18}, {6, 
            10, 16, 14, 2, 18}, {6, 10, 16, 20, 2, 6}, {6, 10, 16, 20, 2, 
            12}, {6, 10, 16, 20, 6, 2}, {6, 10, 20, 14, 2, 14}, {6, 10, 20, 
            16, 2, 6}, {6, 10, 20, 16, 2, 12}, {6, 10, 20, 16, 6, 2}, {6, 10, 
            22, 10, 12, 2}, {6, 10, 22, 14, 8, 2}, {6, 12, 2, 14, 20, 
            10}, {6, 12, 2, 14, 20, 12}, {6, 12, 2, 16, 18, 14}, {6, 12, 10, 
            8, 22, 2}, {6, 12, 10, 22, 10, 2}, {6, 12, 12, 8, 22, 2}, {6, 12, 
            12, 22, 8, 2}, {6, 12, 14, 20, 6, 2}, {6, 12, 16, 18, 2, 6}, {6, 
            12, 16, 20, 2, 6}, {6, 12, 16, 20, 6, 2}, {6, 12, 18, 16, 2, 
            6}, {6, 12, 20, 16, 2, 6}, {6, 12, 20, 16, 6, 2}, {6, 14, 2, 6, 
            16, 16}, {6, 14, 2, 6, 16, 18}, {6, 14, 2, 6, 20, 12}, {6, 14, 2, 
            6, 20, 14}, {6, 14, 2, 10, 16, 18}, {6, 14, 2, 10, 20, 14}, {6, 
            14, 2, 14, 20, 8}, {6, 14, 2, 14, 20, 10}, {6, 14, 2, 16, 18, 
            8}, {6, 14, 2, 16, 18, 12}, {6, 14, 8, 8, 22, 2}, {6, 14, 8, 10, 
            22, 2}, {6, 14, 10, 8, 22, 2}, {6, 14, 12, 6, 20, 2}, {6, 14, 14, 
            6, 20, 2}, {6, 14, 14, 20, 6, 2}, {6, 14, 16, 6, 16, 2}, {6, 14, 
            16, 18, 2, 6}, {6, 14, 18, 6, 16, 2}, {6, 14, 18, 16, 2, 6}, {6, 
            14, 20, 12, 2, 10}, {6, 14, 20, 12, 2, 12}, {6, 14, 20, 14, 2, 
            8}, {6, 14, 20, 14, 2, 10}, {6, 16, 2, 6, 20, 10}, {6, 16, 2, 6, 
            20, 12}, {6, 16, 2, 10, 20, 6}, {6, 16, 2, 10, 20, 12}, {6, 16, 
            2, 12, 18, 6}, {6, 16, 2, 12, 20, 6}, {6, 16, 2, 14, 18, 6}, {6, 
            16, 6, 2, 14, 16}, {6, 16, 6, 2, 14, 18}, {6, 16, 6, 2, 20, 
            10}, {6, 16, 6, 2, 20, 12}, {6, 16, 6, 10, 20, 2}, {6, 16, 6, 12, 
            20, 2}, {6, 16, 6, 16, 14, 2}, {6, 16, 6, 18, 14, 2}, {6, 16, 10, 
            2, 10, 18}, {6, 16, 10, 2, 14, 18}, {6, 16, 10, 2, 20, 6}, {6, 
            16, 10, 2, 20, 12}, {6, 16, 10, 6, 20, 2}, {6, 16, 12, 2, 18, 
            6}, {6, 16, 12, 2, 20, 6}, {6, 16, 12, 6, 20, 2}, {6, 16, 14, 2, 
            18, 6}, {6, 16, 14, 16, 6, 2}, {6, 16, 18, 10, 2, 12}, {6, 16, 
            18, 12, 2, 14}, {6, 16, 18, 14, 2, 8}, {6, 16, 18, 14, 2, 
            12}, {6, 18, 2, 12, 16, 6}, {6, 18, 2, 14, 16, 6}, {6, 18, 12, 2, 
            16, 6}, {6, 18, 14, 2, 16, 6}, {6, 18, 14, 16, 6, 2}, {6, 18, 16, 
            2, 10, 12}, {6, 18, 16, 2, 12, 14}, {6, 18, 16, 2, 14, 8}, {6, 
            18, 16, 2, 14, 12}, {6, 20, 2, 6, 16, 10}, {6, 20, 2, 6, 16, 
            12}, {6, 20, 2, 10, 16, 6}, {6, 20, 2, 10, 16, 12}, {6, 20, 2, 
            12, 16, 6}, {6, 20, 6, 2, 14, 12}, {6, 20, 6, 2, 14, 14}, {6, 20, 
            6, 2, 16, 10}, {6, 20, 6, 2, 16, 12}, {6, 20, 6, 10, 16, 2}, {6, 
            20, 6, 12, 14, 2}, {6, 20, 6, 12, 16, 2}, {6, 20, 6, 14, 14, 
            2}, {6, 20, 10, 2, 14, 14}, {6, 20, 10, 2, 16, 6}, {6, 20, 10, 2, 
            16, 12}, {6, 20, 10, 6, 16, 2}, {6, 20, 12, 2, 16, 6}, {6, 20, 
            12, 6, 16, 2}, {6, 20, 14, 2, 12, 10}, {6, 20, 14, 2, 12, 
            12}, {6, 20, 14, 2, 14, 8}, {6, 20, 14, 2, 14, 10}, {6, 22, 8, 8, 
            14, 2}, {6, 22, 8, 10, 12, 2}, {6, 22, 8, 10, 14, 2}, {6, 22, 8, 
            12, 12, 2}, {6, 22, 10, 8, 14, 2}, {6, 22, 10, 12, 10, 2}, {8, 2, 
            8, 14, 10, 18}, {8, 2, 8, 14, 16, 18}, {8, 2, 8, 18, 10, 14}, {8, 
            2, 8, 18, 16, 14}, {8, 2, 10, 18, 16, 14}, {8, 2, 10, 20, 14, 
            12}, {8, 2, 12, 22, 6, 10}, {8, 2, 12, 22, 6, 12}, {8, 2, 12, 22, 
            10, 10}, {8, 2, 12, 22, 10, 12}, {8, 2, 14, 18, 16, 8}, {8, 2, 
            14, 18, 16, 10}, {8, 2, 14, 20, 14, 8}, {8, 2, 14, 22, 6, 8}, {8, 
            2, 14, 22, 6, 10}, {8, 2, 14, 22, 10, 8}, {8, 2, 14, 22, 10, 
            10}, {8, 2, 18, 14, 16, 6}, {8, 2, 18, 14, 16, 8}, {8, 2, 20, 14, 
            14, 6}, {8, 2, 20, 14, 14, 8}, {8, 2, 22, 14, 6, 8}, {8, 2, 22, 
            14, 6, 10}, {8, 2, 22, 14, 10, 8}, {8, 2, 22, 14, 10, 10}, {8, 6, 
            14, 22, 2, 8}, {8, 6, 14, 22, 2, 10}, {8, 6, 22, 12, 2, 10}, {8, 
            6, 22, 12, 2, 12}, {8, 6, 22, 14, 2, 8}, {8, 6, 22, 14, 2, 
            10}, {8, 8, 2, 10, 14, 18}, {8, 8, 2, 10, 18, 14}, {8, 8, 2, 16, 
            14, 18}, {8, 8, 2, 16, 18, 14}, {8, 10, 2, 14, 20, 12}, {8, 10, 
            2, 16, 18, 14}, {8, 10, 14, 8, 2, 18}, {8, 10, 14, 22, 2, 8}, {8, 
            10, 14, 22, 2, 10}, {8, 10, 18, 8, 2, 14}, {8, 10, 22, 12, 2, 
            10}, {8, 10, 22, 12, 2, 12}, {8, 10, 22, 14, 2, 8}, {8, 10, 22, 
            14, 2, 10}, {8, 12, 2, 6, 22, 10}, {8, 12, 2, 6, 22, 12}, {8, 12, 
            2, 10, 22, 10}, {8, 12, 2, 10, 22, 12}, {8, 14, 2, 6, 22, 8}, {8, 
            14, 2, 6, 22, 10}, {8, 14, 2, 10, 22, 8}, {8, 14, 2, 10, 22, 
            10}, {8, 14, 2, 14, 20, 8}, {8, 14, 2, 16, 18, 8}, {8, 14, 2, 16, 
            18, 10}, {8, 14, 6, 2, 22, 8}, {8, 14, 6, 2, 22, 10}, {8, 14, 10, 
            2, 8, 18}, {8, 14, 10, 2, 22, 8}, {8, 14, 10, 2, 22, 10}, {8, 14, 
            14, 2, 20, 6}, {8, 14, 14, 2, 20, 8}, {8, 14, 14, 20, 2, 6}, {8, 
            14, 14, 20, 2, 8}, {8, 14, 16, 2, 8, 18}, {8, 14, 16, 2, 18, 
            6}, {8, 14, 16, 2, 18, 8}, {8, 14, 20, 10, 2, 12}, {8, 14, 20, 
            14, 2, 8}, {8, 16, 14, 8, 2, 18}, {8, 16, 14, 18, 2, 6}, {8, 16, 
            14, 18, 2, 8}, {8, 16, 18, 8, 2, 14}, {8, 16, 18, 10, 2, 14}, {8, 
            16, 18, 14, 2, 8}, {8, 16, 18, 14, 2, 10}, {8, 18, 2, 16, 14, 
            6}, {8, 18, 2, 16, 14, 8}, {8, 18, 10, 2, 8, 14}, {8, 18, 16, 2, 
            8, 14}, {8, 18, 16, 2, 10, 14}, {8, 18, 16, 2, 14, 8}, {8, 18, 
            16, 2, 14, 10}, {8, 20, 2, 14, 14, 6}, {8, 20, 2, 14, 14, 8}, {8, 
            20, 14, 2, 10, 12}, {8, 20, 14, 2, 14, 8}, {8, 22, 2, 6, 14, 
            8}, {8, 22, 2, 6, 14, 10}, {8, 22, 2, 10, 14, 8}, {8, 22, 2, 10, 
            14, 10}, {8, 22, 6, 2, 12, 10}, {8, 22, 6, 2, 12, 12}, {8, 22, 6, 
            2, 14, 8}, {8, 22, 6, 2, 14, 10}, {8, 22, 10, 2, 12, 10}, {8, 22, 
            10, 2, 12, 12}, {8, 22, 10, 2, 14, 8}, {8, 22, 10, 2, 14, 
            10}, {10, 2, 6, 12, 16, 18}, {10, 2, 6, 18, 10, 16}, {10, 2, 8, 
            12, 14, 20}, {10, 2, 8, 14, 16, 18}, {10, 2, 10, 12, 6, 22}, {10, 
            2, 10, 12, 10, 22}, {10, 2, 10, 22, 6, 12}, {10, 2, 10, 22, 10, 
            12}, {10, 2, 14, 22, 6, 8}, {10, 2, 14, 22, 10, 8}, {10, 2, 16, 
            20, 6, 6}, {10, 2, 18, 14, 16, 8}, {10, 2, 20, 12, 14, 6}, {10, 
            2, 20, 14, 14, 6}, {10, 2, 20, 16, 6, 6}, {10, 2, 22, 12, 6, 
            8}, {10, 2, 22, 12, 10, 8}, {10, 2, 22, 14, 6, 8}, {10, 2, 22, 
            14, 10, 8}, {10, 6, 2, 10, 18, 16}, {10, 6, 2, 16, 12, 18}, {10, 
            6, 12, 10, 2, 22}, {10, 6, 12, 22, 2, 8}, {10, 6, 14, 22, 2, 
            8}, {10, 6, 16, 10, 18, 2}, {10, 6, 16, 14, 18, 2}, {10, 6, 16, 
            20, 2, 6}, {10, 6, 16, 20, 6, 2}, {10, 6, 16, 20, 12, 2}, {10, 6, 
            20, 14, 14, 2}, {10, 6, 20, 16, 2, 6}, {10, 6, 20, 16, 6, 
            2}, {10, 6, 20, 16, 12, 2}, {10, 6, 22, 10, 2, 12}, {10, 6, 22, 
            14, 2, 8}, {10, 8, 2, 14, 12, 20}, {10, 8, 2, 16, 14, 18}, {10, 
            8, 14, 8, 18, 2}, {10, 8, 14, 22, 8, 2}, {10, 8, 14, 22, 10, 
            2}, {10, 8, 18, 8, 14, 2}, {10, 8, 22, 12, 10, 2}, {10, 8, 22, 
            12, 12, 2}, {10, 8, 22, 14, 8, 2}, {10, 8, 22, 14, 10, 2}, {10, 
            10, 2, 6, 12, 22}, {10, 10, 2, 6, 22, 12}, {10, 10, 2, 10, 12, 
            22}, {10, 10, 2, 10, 22, 12}, {10, 10, 12, 10, 2, 22}, {10, 10, 
            12, 10, 22, 2}, {10, 10, 12, 22, 2, 8}, {10, 10, 12, 22, 8, 
            2}, {10, 10, 14, 22, 2, 8}, {10, 10, 14, 22, 8, 2}, {10, 10, 18, 
            6, 2, 16}, {10, 10, 18, 6, 16, 2}, {10, 10, 22, 10, 2, 12}, {10, 
            10, 22, 10, 12, 2}, {10, 10, 22, 14, 2, 8}, {10, 10, 22, 14, 8, 
            2}, {10, 12, 6, 2, 10, 22}, {10, 12, 6, 2, 22, 8}, {10, 12, 10, 
            2, 10, 22}, {10, 12, 10, 2, 22, 8}, {10, 12, 10, 8, 22, 2}, {10, 
            12, 10, 22, 10, 2}, {10, 12, 12, 8, 22, 2}, {10, 12, 12, 22, 8, 
            2}, {10, 12, 14, 2, 8, 20}, {10, 12, 14, 2, 20, 6}, {10, 12, 16, 
            2, 6, 18}, {10, 12, 16, 20, 6, 2}, {10, 12, 20, 16, 6, 2}, {10, 
            14, 2, 6, 22, 8}, {10, 14, 2, 10, 22, 8}, {10, 14, 6, 2, 22, 
            8}, {10, 14, 8, 8, 22, 2}, {10, 14, 8, 10, 22, 2}, {10, 14, 8, 
            18, 8, 2}, {10, 14, 10, 2, 22, 8}, {10, 14, 10, 8, 22, 2}, {10, 
            14, 12, 8, 2, 20}, {10, 14, 12, 20, 2, 6}, {10, 14, 14, 2, 20, 
            6}, {10, 14, 14, 6, 20, 2}, {10, 14, 14, 20, 2, 6}, {10, 14, 14, 
            20, 6, 2}, {10, 14, 16, 2, 8, 18}, {10, 14, 16, 2, 18, 8}, {10, 
            14, 18, 6, 16, 2}, {10, 16, 2, 6, 20, 6}, {10, 16, 6, 2, 20, 
            6}, {10, 16, 6, 6, 20, 2}, {10, 16, 6, 12, 20, 2}, {10, 16, 6, 
            18, 10, 2}, {10, 16, 6, 18, 14, 2}, {10, 16, 12, 6, 2, 18}, {10, 
            16, 12, 6, 20, 2}, {10, 16, 14, 8, 2, 18}, {10, 16, 14, 18, 2, 
            8}, {10, 18, 2, 16, 14, 8}, {10, 18, 8, 14, 8, 2}, {10, 18, 10, 
            2, 6, 16}, {10, 18, 10, 16, 6, 2}, {10, 18, 14, 16, 6, 2}, {10, 
            20, 2, 6, 16, 6}, {10, 20, 2, 14, 12, 6}, {10, 20, 2, 14, 14, 
            6}, {10, 20, 6, 2, 16, 6}, {10, 20, 6, 6, 16, 2}, {10, 20, 6, 12, 
            16, 2}, {10, 20, 6, 14, 14, 2}, {10, 20, 12, 6, 16, 2}, {10, 22, 
            2, 6, 12, 8}, {10, 22, 2, 6, 14, 8}, {10, 22, 2, 10, 12, 8}, {10, 
            22, 2, 10, 14, 8}, {10, 22, 6, 2, 10, 12}, {10, 22, 6, 2, 14, 
            8}, {10, 22, 8, 8, 14, 2}, {10, 22, 8, 10, 12, 2}, {10, 22, 8, 
            10, 14, 2}, {10, 22, 8, 12, 12, 2}, {10, 22, 10, 2, 10, 12}, {10, 
            22, 10, 2, 14, 8}, {10, 22, 10, 8, 14, 2}, {10, 22, 10, 12, 10, 
            2}, {12, 2, 6, 10, 14, 20}, {12, 2, 6, 12, 14, 20}, {12, 2, 6, 
            14, 16, 18}, {12, 2, 8, 10, 6, 22}, {12, 2, 8, 10, 10, 22}, {12, 
            2, 8, 12, 6, 22}, {12, 2, 8, 12, 10, 22}, {12, 2, 16, 20, 6, 
            6}, {12, 2, 16, 20, 10, 6}, {12, 2, 18, 10, 16, 6}, {12, 2, 18, 
            14, 16, 6}, {12, 2, 20, 10, 14, 8}, {12, 2, 20, 12, 14, 6}, {12, 
            2, 20, 14, 6, 6}, {12, 2, 20, 16, 6, 6}, {12, 2, 20, 16, 10, 
            6}, {12, 2, 22, 10, 6, 10}, {12, 2, 22, 10, 10, 10}, {12, 2, 22, 
            12, 6, 8}, {12, 2, 22, 12, 10, 8}, {12, 6, 2, 14, 10, 20}, {12, 
            6, 2, 14, 12, 20}, {12, 6, 2, 16, 14, 18}, {12, 6, 10, 8, 2, 
            22}, {12, 6, 10, 22, 2, 10}, {12, 6, 12, 8, 2, 22}, {12, 6, 12, 
            22, 2, 8}, {12, 6, 14, 20, 2, 6}, {12, 6, 16, 18, 6, 2}, {12, 6, 
            16, 20, 2, 6}, {12, 6, 16, 20, 6, 2}, {12, 6, 18, 16, 6, 2}, {12, 
            6, 20, 16, 2, 6}, {12, 6, 20, 16, 6, 2}, {12, 8, 2, 6, 10, 
            22}, {12, 8, 2, 6, 12, 22}, {12, 8, 2, 10, 10, 22}, {12, 8, 2, 
            10, 12, 22}, {12, 10, 6, 2, 8, 22}, {12, 10, 6, 2, 22, 10}, {12, 
            10, 10, 2, 8, 22}, {12, 10, 10, 2, 22, 10}, {12, 10, 10, 8, 2, 
            22}, {12, 10, 10, 22, 2, 10}, {12, 10, 12, 8, 2, 22}, {12, 10, 
            12, 22, 2, 8}, {12, 10, 14, 2, 6, 20}, {12, 10, 14, 2, 20, 
            8}, {12, 10, 16, 2, 18, 6}, {12, 10, 16, 20, 2, 6}, {12, 10, 20, 
            16, 2, 6}, {12, 12, 6, 2, 8, 22}, {12, 12, 6, 2, 22, 8}, {12, 12, 
            10, 2, 8, 22}, {12, 12, 10, 2, 22, 8}, {12, 12, 14, 2, 6, 
            20}, {12, 12, 14, 2, 20, 6}, {12, 14, 6, 2, 20, 6}, {12, 14, 10, 
            6, 2, 20}, {12, 14, 10, 20, 2, 8}, {12, 14, 12, 6, 2, 20}, {12, 
            14, 12, 20, 2, 6}, {12, 14, 16, 2, 6, 18}, {12, 14, 16, 2, 18, 
            6}, {12, 16, 2, 6, 20, 6}, {12, 16, 2, 10, 20, 6}, {12, 16, 6, 2, 
            20, 6}, {12, 16, 6, 6, 18, 2}, {12, 16, 6, 6, 20, 2}, {12, 16, 
            10, 2, 20, 6}, {12, 16, 10, 18, 2, 6}, {12, 16, 14, 6, 2, 
            18}, {12, 16, 14, 18, 2, 6}, {12, 18, 2, 16, 10, 6}, {12, 18, 2, 
            16, 14, 6}, {12, 18, 6, 6, 16, 2}, {12, 20, 2, 6, 14, 6}, {12, 
            20, 2, 6, 16, 6}, {12, 20, 2, 10, 16, 6}, {12, 20, 2, 14, 10, 
            8}, {12, 20, 2, 14, 12, 6}, {12, 20, 6, 2, 16, 6}, {12, 20, 6, 6, 
            16, 2}, {12, 20, 10, 2, 16, 6}, {12, 22, 2, 6, 10, 10}, {12, 22, 
            2, 6, 12, 8}, {12, 22, 2, 10, 10, 10}, {12, 22, 2, 10, 12, 
            8}, {14, 2, 6, 8, 14, 20}, {14, 2, 6, 8, 16, 18}, {14, 2, 6, 10, 
            14, 20}, {14, 2, 6, 12, 6, 20}, {14, 2, 6, 12, 16, 18}, {14, 2, 
            6, 14, 6, 20}, {14, 2, 6, 14, 10, 20}, {14, 2, 6, 16, 6, 
            16}, {14, 2, 6, 18, 6, 16}, {14, 2, 6, 18, 10, 16}, {14, 2, 8, 8, 
            6, 22}, {14, 2, 8, 8, 10, 22}, {14, 2, 8, 8, 14, 20}, {14, 2, 8, 
            8, 16, 18}, {14, 2, 8, 10, 6, 22}, {14, 2, 8, 10, 10, 22}, {14, 
            2, 8, 10, 16, 18}, {14, 2, 10, 8, 6, 22}, {14, 2, 10, 8, 10, 
            22}, {14, 2, 18, 8, 10, 8}, {14, 2, 18, 8, 16, 8}, {14, 2, 18, 
            10, 16, 8}, {14, 2, 18, 12, 16, 6}, {14, 2, 20, 14, 6, 6}, {14, 
            2, 20, 14, 10, 6}, {14, 6, 2, 6, 12, 20}, {14, 6, 2, 6, 14, 
            20}, {14, 6, 2, 6, 16, 16}, {14, 6, 2, 6, 18, 16}, {14, 6, 2, 10, 
            14, 20}, {14, 6, 2, 10, 18, 16}, {14, 6, 2, 14, 8, 20}, {14, 6, 
            2, 14, 10, 20}, {14, 6, 2, 16, 8, 18}, {14, 6, 2, 16, 12, 
            18}, {14, 6, 8, 8, 2, 22}, {14, 6, 8, 10, 2, 22}, {14, 6, 10, 8, 
            2, 22}, {14, 6, 12, 6, 2, 20}, {14, 6, 14, 6, 2, 20}, {14, 6, 14, 
            20, 2, 6}, {14, 6, 16, 6, 2, 16}, {14, 6, 16, 18, 6, 2}, {14, 6, 
            18, 6, 2, 16}, {14, 6, 18, 16, 6, 2}, {14, 6, 20, 12, 10, 
            2}, {14, 6, 20, 12, 12, 2}, {14, 6, 20, 14, 8, 2}, {14, 6, 20, 
            14, 10, 2}, {14, 8, 2, 6, 8, 22}, {14, 8, 2, 6, 10, 22}, {14, 8, 
            2, 10, 8, 22}, {14, 8, 2, 10, 10, 22}, {14, 8, 2, 14, 8, 
            20}, {14, 8, 2, 16, 8, 18}, {14, 8, 2, 16, 10, 18}, {14, 8, 6, 2, 
            8, 22}, {14, 8, 6, 2, 10, 22}, {14, 8, 10, 2, 8, 22}, {14, 8, 10, 
            2, 10, 22}, {14, 8, 10, 2, 18, 8}, {14, 8, 14, 2, 6, 20}, {14, 8, 
            14, 2, 8, 20}, {14, 8, 14, 20, 6, 2}, {14, 8, 14, 20, 8, 2}, {14, 
            8, 16, 2, 6, 18}, {14, 8, 16, 2, 8, 18}, {14, 8, 16, 2, 18, 
            8}, {14, 8, 20, 10, 12, 2}, {14, 8, 20, 14, 8, 2}, {14, 10, 2, 6, 
            8, 22}, {14, 10, 2, 10, 8, 22}, {14, 10, 6, 2, 8, 22}, {14, 10, 
            8, 8, 2, 22}, {14, 10, 8, 10, 2, 22}, {14, 10, 8, 18, 2, 8}, {14, 
            10, 10, 2, 8, 22}, {14, 10, 10, 8, 2, 22}, {14, 10, 12, 8, 20, 
            2}, {14, 10, 12, 20, 6, 2}, {14, 10, 14, 2, 6, 20}, {14, 10, 14, 
            6, 2, 20}, {14, 10, 14, 20, 2, 6}, {14, 10, 14, 20, 6, 2}, {14, 
            10, 16, 2, 8, 18}, {14, 10, 16, 2, 18, 8}, {14, 10, 18, 6, 2, 
            16}, {14, 12, 6, 2, 6, 20}, {14, 12, 10, 6, 20, 2}, {14, 12, 10, 
            20, 8, 2}, {14, 12, 12, 6, 20, 2}, {14, 12, 12, 20, 6, 2}, {14, 
            12, 16, 2, 6, 18}, {14, 12, 16, 2, 18, 6}, {14, 14, 6, 2, 6, 
            20}, {14, 14, 6, 2, 20, 6}, {14, 14, 8, 6, 2, 20}, {14, 14, 8, 6, 
            20, 2}, {14, 14, 8, 8, 2, 20}, {14, 14, 8, 8, 20, 2}, {14, 14, 
            10, 2, 6, 20}, {14, 14, 10, 2, 20, 6}, {14, 14, 10, 6, 2, 
            20}, {14, 14, 10, 6, 20, 2}, {14, 16, 6, 2, 6, 16}, {14, 16, 6, 
            6, 18, 2}, {14, 16, 8, 6, 2, 18}, {14, 16, 8, 8, 2, 18}, {14, 16, 
            8, 18, 2, 8}, {14, 16, 10, 8, 2, 18}, {14, 16, 10, 18, 2, 
            8}, {14, 16, 12, 6, 2, 18}, {14, 16, 12, 18, 2, 6}, {14, 18, 2, 
            10, 8, 8}, {14, 18, 2, 16, 8, 8}, {14, 18, 2, 16, 10, 8}, {14, 
            18, 2, 16, 12, 6}, {14, 18, 6, 2, 6, 16}, {14, 18, 6, 6, 16, 
            2}, {14, 18, 10, 2, 6, 16}, {14, 20, 2, 6, 14, 6}, {14, 20, 2, 
            10, 14, 6}, {14, 20, 6, 8, 14, 2}, {14, 20, 6, 10, 12, 2}, {14, 
            20, 6, 10, 14, 2}, {14, 20, 6, 12, 12, 2}, {14, 20, 8, 8, 14, 
            2}, {14, 20, 8, 12, 10, 2}, {16, 2, 6, 6, 10, 20}, {16, 2, 6, 6, 
            12, 18}, {16, 2, 6, 6, 12, 20}, {16, 2, 6, 6, 14, 18}, {16, 2, 6, 
            10, 6, 20}, {16, 2, 6, 12, 6, 20}, {16, 2, 6, 12, 10, 20}, {16, 
            2, 10, 6, 6, 20}, {16, 2, 12, 6, 6, 20}, {16, 2, 12, 6, 10, 
            20}, {16, 2, 16, 6, 6, 14}, {16, 2, 16, 14, 6, 6}, {16, 2, 18, 6, 
            6, 14}, {16, 2, 18, 6, 10, 10}, {16, 2, 18, 6, 10, 14}, {16, 6, 
            2, 6, 10, 20}, {16, 6, 2, 6, 12, 20}, {16, 6, 2, 10, 6, 20}, {16, 
            6, 2, 10, 12, 20}, {16, 6, 2, 12, 6, 18}, {16, 6, 2, 12, 6, 
            20}, {16, 6, 2, 14, 6, 18}, {16, 6, 6, 2, 10, 20}, {16, 6, 6, 2, 
            12, 20}, {16, 6, 6, 2, 16, 14}, {16, 6, 6, 2, 18, 14}, {16, 6, 6, 
            10, 2, 20}, {16, 6, 6, 12, 2, 20}, {16, 6, 6, 16, 2, 14}, {16, 6, 
            6, 18, 2, 14}, {16, 6, 10, 2, 6, 20}, {16, 6, 10, 2, 12, 
            20}, {16, 6, 10, 2, 18, 10}, {16, 6, 10, 2, 18, 14}, {16, 6, 10, 
            6, 2, 20}, {16, 6, 12, 2, 6, 18}, {16, 6, 12, 2, 6, 20}, {16, 6, 
            12, 6, 2, 20}, {16, 6, 14, 2, 6, 18}, {16, 6, 14, 16, 2, 6}, {16, 
            6, 18, 10, 12, 2}, {16, 6, 18, 12, 14, 2}, {16, 6, 18, 14, 8, 
            2}, {16, 6, 18, 14, 12, 2}, {16, 8, 14, 8, 18, 2}, {16, 8, 14, 
            18, 6, 2}, {16, 8, 14, 18, 8, 2}, {16, 8, 18, 8, 14, 2}, {16, 8, 
            18, 10, 14, 2}, {16, 8, 18, 14, 8, 2}, {16, 8, 18, 14, 10, 
            2}, {16, 10, 2, 6, 6, 20}, {16, 10, 6, 2, 6, 20}, {16, 10, 6, 6, 
            2, 20}, {16, 10, 6, 12, 2, 20}, {16, 10, 6, 18, 2, 10}, {16, 10, 
            6, 18, 2, 14}, {16, 10, 12, 6, 2, 20}, {16, 10, 12, 6, 18, 
            2}, {16, 10, 14, 8, 18, 2}, {16, 10, 14, 18, 8, 2}, {16, 12, 2, 
            6, 6, 20}, {16, 12, 2, 10, 6, 20}, {16, 12, 6, 2, 6, 20}, {16, 
            12, 6, 6, 2, 18}, {16, 12, 6, 6, 2, 20}, {16, 12, 10, 2, 6, 
            20}, {16, 12, 10, 18, 6, 2}, {16, 12, 14, 6, 18, 2}, {16, 12, 14, 
            18, 6, 2}, {16, 14, 6, 2, 16, 6}, {16, 14, 6, 6, 2, 18}, {16, 14, 
            8, 6, 18, 2}, {16, 14, 8, 8, 18, 2}, {16, 14, 8, 18, 8, 2}, {16, 
            14, 10, 8, 18, 2}, {16, 14, 10, 18, 8, 2}, {16, 14, 12, 6, 18, 
            2}, {16, 14, 12, 18, 6, 2}, {16, 16, 2, 6, 6, 14}, {16, 16, 2, 6, 
            14, 6}, {16, 18, 2, 6, 6, 14}, {16, 18, 2, 10, 6, 10}, {16, 18, 
            2, 10, 6, 14}, {16, 18, 6, 8, 14, 2}, {16, 18, 6, 12, 10, 
            2}, {16, 18, 6, 12, 14, 2}, {16, 18, 6, 14, 12, 2}, {16, 18, 8, 
            8, 14, 2}, {16, 18, 8, 10, 14, 2}, {16, 18, 8, 14, 8, 2}, {16, 
            18, 8, 14, 10, 2}, {18, 2, 6, 6, 12, 16}, {18, 2, 6, 6, 14, 
            16}, {18, 2, 8, 6, 16, 14}, {18, 2, 8, 8, 16, 14}, {18, 2, 10, 8, 
            16, 14}, {18, 2, 12, 6, 16, 10}, {18, 2, 12, 6, 16, 14}, {18, 2, 
            14, 6, 16, 12}, {18, 2, 14, 8, 10, 8}, {18, 2, 14, 8, 16, 
            8}, {18, 2, 14, 8, 16, 10}, {18, 2, 16, 10, 10, 6}, {18, 2, 16, 
            14, 6, 6}, {18, 2, 16, 14, 10, 6}, {18, 6, 2, 12, 6, 16}, {18, 6, 
            2, 14, 6, 16}, {18, 6, 12, 2, 6, 16}, {18, 6, 14, 2, 6, 16}, {18, 
            6, 14, 16, 2, 6}, {18, 6, 16, 2, 8, 14}, {18, 6, 16, 2, 12, 
            10}, {18, 6, 16, 2, 12, 14}, {18, 6, 16, 2, 14, 12}, {18, 8, 2, 
            16, 6, 14}, {18, 8, 2, 16, 8, 14}, {18, 8, 10, 2, 14, 8}, {18, 8, 
            16, 2, 8, 14}, {18, 8, 16, 2, 10, 14}, {18, 8, 16, 2, 14, 
            8}, {18, 8, 16, 2, 14, 10}, {18, 10, 2, 16, 8, 14}, {18, 10, 8, 
            14, 2, 8}, {18, 10, 10, 2, 16, 6}, {18, 10, 10, 16, 2, 6}, {18, 
            10, 14, 16, 2, 6}, {18, 12, 2, 16, 6, 10}, {18, 12, 2, 16, 6, 
            14}, {18, 12, 6, 6, 2, 16}, {18, 14, 2, 10, 8, 8}, {18, 14, 2, 
            16, 6, 12}, {18, 14, 2, 16, 8, 8}, {18, 14, 2, 16, 8, 10}, {18, 
            14, 6, 2, 16, 6}, {18, 14, 6, 6, 2, 16}, {18, 14, 10, 2, 16, 
            6}, {18, 16, 2, 6, 14, 6}, {18, 16, 2, 10, 10, 6}, {18, 16, 2, 
            10, 14, 6}, {18, 16, 6, 8, 2, 14}, {18, 16, 6, 12, 2, 10}, {18, 
            16, 6, 12, 2, 14}, {18, 16, 6, 14, 2, 12}, {18, 16, 8, 8, 2, 
            14}, {18, 16, 8, 10, 2, 14}, {18, 16, 8, 14, 2, 8}, {18, 16, 8, 
            14, 2, 10}, {20, 2, 6, 6, 10, 16}, {20, 2, 6, 6, 12, 16}, {20, 2, 
            6, 10, 6, 16}, {20, 2, 6, 12, 6, 16}, {20, 2, 6, 12, 10, 
            16}, {20, 2, 8, 6, 14, 14}, {20, 2, 8, 8, 14, 14}, {20, 2, 10, 6, 
            6, 16}, {20, 2, 10, 6, 14, 12}, {20, 2, 10, 6, 14, 14}, {20, 2, 
            12, 6, 6, 14}, {20, 2, 12, 6, 6, 16}, {20, 2, 12, 6, 10, 
            16}, {20, 2, 12, 6, 14, 12}, {20, 2, 12, 8, 14, 10}, {20, 2, 14, 
            6, 6, 14}, {20, 2, 14, 6, 10, 14}, {20, 6, 2, 6, 10, 16}, {20, 6, 
            2, 6, 12, 16}, {20, 6, 2, 10, 6, 16}, {20, 6, 2, 10, 12, 
            16}, {20, 6, 2, 12, 6, 16}, {20, 6, 6, 2, 10, 16}, {20, 6, 6, 2, 
            12, 14}, {20, 6, 6, 2, 12, 16}, {20, 6, 6, 2, 14, 14}, {20, 6, 6, 
            10, 2, 16}, {20, 6, 6, 12, 2, 14}, {20, 6, 6, 12, 2, 16}, {20, 6, 
            6, 14, 2, 14}, {20, 6, 10, 2, 6, 16}, {20, 6, 10, 2, 12, 
            16}, {20, 6, 10, 2, 14, 14}, {20, 6, 10, 6, 2, 16}, {20, 6, 12, 
            2, 6, 16}, {20, 6, 12, 6, 2, 16}, {20, 6, 14, 2, 8, 14}, {20, 6, 
            14, 2, 10, 12}, {20, 6, 14, 2, 10, 14}, {20, 6, 14, 2, 12, 
            12}, {20, 8, 2, 14, 6, 14}, {20, 8, 2, 14, 8, 14}, {20, 8, 14, 2, 
            8, 14}, {20, 8, 14, 2, 12, 10}, {20, 10, 2, 6, 6, 16}, {20, 10, 
            2, 14, 6, 12}, {20, 10, 2, 14, 6, 14}, {20, 10, 6, 2, 6, 
            16}, {20, 10, 6, 6, 2, 16}, {20, 10, 6, 12, 2, 16}, {20, 10, 6, 
            14, 2, 14}, {20, 10, 12, 6, 2, 16}, {20, 12, 2, 6, 6, 14}, {20, 
            12, 2, 6, 6, 16}, {20, 12, 2, 10, 6, 16}, {20, 12, 2, 14, 6, 
            12}, {20, 12, 2, 14, 8, 10}, {20, 12, 6, 2, 6, 16}, {20, 12, 6, 
            6, 2, 16}, {20, 12, 10, 2, 6, 16}, {20, 14, 2, 6, 6, 14}, {20, 
            14, 2, 10, 6, 14}, {20, 14, 6, 8, 2, 14}, {20, 14, 6, 10, 2, 
            12}, {20, 14, 6, 10, 2, 14}, {20, 14, 6, 12, 2, 12}, {20, 14, 8, 
            8, 2, 14}, {20, 14, 8, 12, 2, 10}, {22, 2, 8, 8, 6, 14}, {22, 2, 
            8, 8, 10, 14}, {22, 2, 8, 10, 6, 14}, {22, 2, 8, 10, 10, 
            14}, {22, 2, 10, 8, 6, 12}, {22, 2, 10, 8, 6, 14}, {22, 2, 10, 8, 
            10, 12}, {22, 2, 10, 8, 10, 14}, {22, 2, 12, 8, 6, 12}, {22, 2, 
            12, 8, 10, 12}, {22, 2, 12, 10, 6, 10}, {22, 2, 12, 10, 10, 
            10}, {22, 6, 8, 8, 2, 14}, {22, 6, 8, 10, 2, 12}, {22, 6, 8, 10, 
            2, 14}, {22, 6, 8, 12, 2, 12}, {22, 6, 10, 8, 2, 14}, {22, 6, 10, 
            12, 2, 10}, {22, 8, 2, 6, 8, 14}, {22, 8, 2, 6, 10, 14}, {22, 8, 
            2, 10, 8, 14}, {22, 8, 2, 10, 10, 14}, {22, 8, 6, 2, 8, 14}, {22, 
            8, 6, 2, 10, 12}, {22, 8, 6, 2, 10, 14}, {22, 8, 6, 2, 12, 
            12}, {22, 8, 10, 2, 8, 14}, {22, 8, 10, 2, 10, 12}, {22, 8, 10, 
            2, 10, 14}, {22, 8, 10, 2, 12, 12}, {22, 10, 2, 6, 8, 12}, {22, 
            10, 2, 6, 8, 14}, {22, 10, 2, 10, 8, 12}, {22, 10, 2, 10, 8, 
            14}, {22, 10, 6, 2, 8, 14}, {22, 10, 6, 2, 12, 10}, {22, 10, 8, 
            8, 2, 14}, {22, 10, 8, 10, 2, 12}, {22, 10, 8, 10, 2, 14}, {22, 
            10, 8, 12, 2, 12}, {22, 10, 10, 2, 8, 14}, {22, 10, 10, 2, 12, 
            10}, {22, 10, 10, 8, 2, 14}, {22, 10, 10, 12, 2, 10}, {22, 12, 2, 
            6, 8, 12}, {22, 12, 2, 6, 10, 10}, {22, 12, 2, 10, 8, 12}, {22, 
            12, 2, 10, 10, 10}, {2, 4, 12, 18, 12, 16}, {2, 4, 12, 18, 14, 
            10}, {2, 4, 12, 18, 14, 16}, {2, 4, 12, 18, 16, 10}, {2, 4, 14, 
            18, 10, 16}, {2, 4, 14, 18, 12, 16}, {2, 4, 14, 18, 14, 8}, {2, 
            4, 14, 18, 16, 8}, {2, 4, 16, 18, 8, 12}, {2, 4, 16, 18, 8, 
            14}, {2, 4, 18, 18, 8, 10}, {2, 4, 18, 18, 8, 12}, {2, 8, 18, 16, 
            4, 12}, {2, 8, 18, 16, 4, 14}, {2, 8, 18, 18, 4, 10}, {2, 8, 18, 
            18, 4, 12}, {2, 10, 18, 14, 4, 16}, {2, 12, 4, 12, 18, 16}, {2, 
            12, 4, 14, 18, 10}, {2, 12, 4, 14, 18, 16}, {2, 12, 4, 16, 18, 
            10}, {2, 12, 18, 12, 4, 16}, {2, 12, 18, 14, 4, 16}, {2, 14, 4, 
            10, 18, 16}, {2, 14, 4, 12, 18, 16}, {2, 14, 4, 14, 18, 8}, {2, 
            14, 4, 16, 18, 8}, {2, 14, 18, 12, 4, 10}, {2, 14, 18, 12, 4, 
            16}, {2, 14, 18, 14, 4, 8}, {2, 16, 4, 8, 18, 12}, {2, 16, 4, 8, 
            18, 14}, {2, 16, 18, 12, 4, 10}, {2, 16, 18, 14, 4, 8}, {2, 18, 
            4, 8, 18, 10}, {2, 18, 4, 8, 18, 12}, {2, 18, 8, 4, 16, 12}, {2, 
            18, 8, 4, 16, 14}, {2, 18, 8, 4, 18, 10}, {2, 18, 8, 4, 18, 
            12}, {2, 18, 10, 4, 14, 16}, {2, 18, 12, 4, 12, 16}, {2, 18, 12, 
            4, 14, 16}, {2, 18, 14, 4, 12, 10}, {2, 18, 14, 4, 12, 16}, {2, 
            18, 14, 4, 14, 8}, {2, 18, 16, 4, 12, 10}, {2, 18, 16, 4, 14, 
            8}, {4, 2, 12, 18, 10, 14}, {4, 2, 12, 18, 10, 16}, {4, 2, 12, 
            18, 16, 12}, {4, 2, 12, 18, 16, 14}, {4, 2, 14, 18, 8, 14}, {4, 
            2, 14, 18, 8, 16}, {4, 2, 14, 18, 16, 10}, {4, 2, 14, 18, 16, 
            12}, {4, 2, 16, 18, 12, 8}, {4, 2, 16, 18, 14, 8}, {4, 2, 18, 18, 
            10, 8}, {4, 2, 18, 18, 12, 8}, {4, 8, 18, 14, 2, 14}, {4, 8, 18, 
            14, 2, 16}, {4, 10, 18, 12, 2, 14}, {4, 10, 18, 12, 2, 16}, {4, 
            10, 18, 18, 2, 8}, {4, 12, 2, 10, 18, 14}, {4, 12, 2, 10, 18, 
            16}, {4, 12, 2, 16, 18, 12}, {4, 12, 2, 16, 18, 14}, {4, 12, 18, 
            16, 2, 8}, {4, 12, 18, 18, 2, 8}, {4, 14, 2, 8, 18, 14}, {4, 14, 
            2, 8, 18, 16}, {4, 14, 2, 16, 18, 10}, {4, 14, 2, 16, 18, 
            12}, {4, 14, 18, 16, 2, 8}, {4, 16, 2, 12, 18, 8}, {4, 16, 2, 14, 
            18, 8}, {4, 16, 18, 12, 2, 12}, {4, 16, 18, 12, 2, 14}, {4, 16, 
            18, 14, 2, 10}, {4, 16, 18, 14, 2, 12}, {4, 18, 2, 10, 18, 
            8}, {4, 18, 2, 12, 18, 8}, {4, 18, 8, 2, 14, 14}, {4, 18, 8, 2, 
            14, 16}, {4, 18, 10, 2, 12, 14}, {4, 18, 10, 2, 12, 16}, {4, 18, 
            10, 2, 18, 8}, {4, 18, 12, 2, 16, 8}, {4, 18, 12, 2, 18, 8}, {4, 
            18, 14, 2, 16, 8}, {4, 18, 16, 2, 12, 12}, {4, 18, 16, 2, 12, 
            14}, {4, 18, 16, 2, 14, 10}, {4, 18, 16, 2, 14, 12}, {8, 2, 18, 
            16, 12, 4}, {8, 2, 18, 16, 14, 4}, {8, 2, 18, 18, 10, 4}, {8, 2, 
            18, 18, 12, 4}, {8, 4, 18, 14, 14, 2}, {8, 4, 18, 14, 16, 2}, {8, 
            10, 18, 18, 2, 4}, {8, 12, 16, 18, 2, 4}, {8, 12, 18, 18, 2, 
            4}, {8, 14, 14, 4, 18, 2}, {8, 14, 14, 18, 4, 2}, {8, 14, 16, 4, 
            18, 2}, {8, 14, 16, 18, 2, 4}, {8, 16, 12, 2, 18, 4}, {8, 16, 14, 
            2, 18, 4}, {8, 16, 14, 18, 4, 2}, {8, 18, 2, 10, 18, 4}, {8, 18, 
            2, 12, 16, 4}, {8, 18, 2, 12, 18, 4}, {8, 18, 2, 14, 16, 4}, {8, 
            18, 4, 14, 14, 2}, {8, 18, 4, 16, 14, 2}, {8, 18, 10, 2, 18, 
            4}, {8, 18, 12, 2, 18, 4}, {10, 2, 18, 14, 16, 4}, {10, 4, 18, 
            12, 14, 2}, {10, 4, 18, 12, 16, 2}, {10, 4, 18, 18, 8, 2}, {10, 
            8, 18, 18, 4, 2}, {10, 12, 14, 4, 18, 2}, {10, 12, 16, 4, 18, 
            2}, {10, 14, 12, 18, 4, 2}, {10, 14, 16, 2, 18, 4}, {10, 16, 12, 
            18, 4, 2}, {10, 16, 14, 18, 2, 4}, {10, 18, 2, 16, 14, 4}, {10, 
            18, 4, 8, 18, 2}, {10, 18, 4, 14, 12, 2}, {10, 18, 4, 16, 12, 
            2}, {10, 18, 8, 4, 18, 2}, {12, 2, 4, 12, 16, 18}, {12, 2, 4, 14, 
            10, 18}, {12, 2, 4, 14, 16, 18}, {12, 2, 4, 16, 10, 18}, {12, 2, 
            18, 12, 16, 4}, {12, 2, 18, 14, 16, 4}, {12, 4, 2, 10, 14, 
            18}, {12, 4, 2, 10, 16, 18}, {12, 4, 2, 16, 12, 18}, {12, 4, 2, 
            16, 14, 18}, {12, 4, 18, 16, 8, 2}, {12, 4, 18, 18, 8, 2}, {12, 
            8, 16, 18, 4, 2}, {12, 8, 18, 18, 4, 2}, {12, 10, 14, 4, 2, 
            18}, {12, 10, 16, 4, 2, 18}, {12, 12, 16, 2, 4, 18}, {12, 12, 16, 
            2, 18, 4}, {12, 14, 10, 2, 4, 18}, {12, 14, 16, 2, 4, 18}, {12, 
            14, 16, 2, 18, 4}, {12, 16, 8, 4, 18, 2}, {12, 16, 10, 2, 4, 
            18}, {12, 16, 12, 4, 2, 18}, {12, 16, 12, 18, 2, 4}, {12, 16, 14, 
            4, 2, 18}, {12, 16, 14, 18, 2, 4}, {12, 18, 2, 16, 12, 4}, {12, 
            18, 2, 16, 14, 4}, {12, 18, 4, 8, 16, 2}, {12, 18, 4, 8, 18, 
            2}, {12, 18, 8, 4, 18, 2}, {14, 2, 4, 10, 16, 18}, {14, 2, 4, 12, 
            16, 18}, {14, 2, 4, 14, 8, 18}, {14, 2, 4, 16, 8, 18}, {14, 2, 
            18, 12, 10, 4}, {14, 2, 18, 12, 16, 4}, {14, 2, 18, 14, 8, 
            4}, {14, 4, 2, 8, 14, 18}, {14, 4, 2, 8, 16, 18}, {14, 4, 2, 16, 
            10, 18}, {14, 4, 2, 16, 12, 18}, {14, 4, 18, 16, 8, 2}, {14, 8, 
            14, 4, 2, 18}, {14, 8, 14, 18, 2, 4}, {14, 8, 16, 4, 2, 18}, {14, 
            8, 16, 18, 4, 2}, {14, 10, 12, 18, 2, 4}, {14, 10, 16, 2, 4, 
            18}, {14, 12, 10, 2, 18, 4}, {14, 12, 16, 2, 4, 18}, {14, 12, 16, 
            2, 18, 4}, {14, 14, 8, 2, 4, 18}, {14, 14, 8, 2, 18, 4}, {14, 16, 
            8, 2, 4, 18}, {14, 16, 8, 4, 18, 2}, {14, 16, 10, 4, 2, 18}, {14, 
            16, 12, 4, 2, 18}, {14, 16, 12, 18, 2, 4}, {14, 18, 2, 8, 14, 
            4}, {14, 18, 2, 10, 12, 4}, {14, 18, 2, 16, 12, 4}, {14, 18, 4, 
            8, 16, 2}, {16, 2, 4, 8, 12, 18}, {16, 2, 4, 8, 14, 18}, {16, 2, 
            18, 12, 10, 4}, {16, 2, 18, 14, 8, 4}, {16, 4, 2, 12, 8, 
            18}, {16, 4, 2, 14, 8, 18}, {16, 4, 18, 12, 12, 2}, {16, 4, 18, 
            12, 14, 2}, {16, 4, 18, 14, 10, 2}, {16, 4, 18, 14, 12, 2}, {16, 
            8, 12, 2, 4, 18}, {16, 8, 14, 2, 4, 18}, {16, 8, 14, 18, 2, 
            4}, {16, 10, 12, 18, 2, 4}, {16, 10, 14, 18, 4, 2}, {16, 12, 8, 
            4, 2, 18}, {16, 12, 10, 2, 18, 4}, {16, 12, 12, 4, 18, 2}, {16, 
            12, 12, 18, 4, 2}, {16, 12, 14, 4, 18, 2}, {16, 12, 14, 18, 4, 
            2}, {16, 14, 8, 2, 18, 4}, {16, 14, 8, 4, 2, 18}, {16, 14, 10, 4, 
            18, 2}, {16, 14, 12, 4, 18, 2}, {16, 14, 12, 18, 4, 2}, {16, 18, 
            2, 8, 14, 4}, {16, 18, 2, 10, 12, 4}, {16, 18, 4, 10, 14, 
            2}, {16, 18, 4, 12, 12, 2}, {16, 18, 4, 12, 14, 2}, {16, 18, 4, 
            14, 12, 2}, {18, 2, 4, 8, 10, 18}, {18, 2, 4, 8, 12, 18}, {18, 2, 
            8, 4, 10, 18}, {18, 2, 8, 4, 12, 16}, {18, 2, 8, 4, 12, 18}, {18, 
            2, 8, 4, 14, 16}, {18, 2, 10, 4, 16, 14}, {18, 2, 12, 4, 16, 
            12}, {18, 2, 12, 4, 16, 14}, {18, 2, 14, 4, 8, 14}, {18, 2, 14, 
            4, 10, 12}, {18, 2, 14, 4, 16, 12}, {18, 2, 16, 4, 8, 14}, {18, 
            2, 16, 4, 10, 12}, {18, 4, 2, 10, 8, 18}, {18, 4, 2, 12, 8, 
            18}, {18, 4, 8, 2, 14, 14}, {18, 4, 8, 2, 16, 14}, {18, 4, 10, 2, 
            8, 18}, {18, 4, 10, 2, 14, 12}, {18, 4, 10, 2, 16, 12}, {18, 4, 
            12, 2, 8, 16}, {18, 4, 12, 2, 8, 18}, {18, 4, 14, 2, 8, 16}, {18, 
            4, 16, 2, 10, 14}, {18, 4, 16, 2, 12, 12}, {18, 4, 16, 2, 12, 
            14}, {18, 4, 16, 2, 14, 12}, {18, 8, 2, 10, 4, 18}, {18, 8, 2, 
            12, 4, 16}, {18, 8, 2, 12, 4, 18}, {18, 8, 2, 14, 4, 16}, {18, 8, 
            4, 14, 2, 14}, {18, 8, 4, 16, 2, 14}, {18, 8, 10, 2, 4, 18}, {18, 
            8, 12, 2, 4, 18}, {18, 10, 2, 16, 4, 14}, {18, 10, 4, 8, 2, 
            18}, {18, 10, 4, 14, 2, 12}, {18, 10, 4, 16, 2, 12}, {18, 10, 8, 
            4, 2, 18}, {18, 12, 2, 16, 4, 12}, {18, 12, 2, 16, 4, 14}, {18, 
            12, 4, 8, 2, 16}, {18, 12, 4, 8, 2, 18}, {18, 12, 8, 4, 2, 
            18}, {18, 14, 2, 8, 4, 14}, {18, 14, 2, 10, 4, 12}, {18, 14, 2, 
            16, 4, 12}, {18, 14, 4, 8, 2, 16}, {18, 16, 2, 8, 4, 14}, {18, 
            16, 2, 10, 4, 12}, {18, 16, 4, 10, 2, 14}, {18, 16, 4, 12, 2, 
            12}, {18, 16, 4, 12, 2, 14}, {18, 16, 4, 14, 2, 
            12}};

    
v1t9bIeq = {-x1 - x2 - x3 - x4 - x5 - 
              x6 ≤ -60, -2 x1 - x2 - x3 - x4 - 
              x5 ≤ -50, -x1 - 2 x2 - x3 - x4 - 
              x6 ≤ -50, -x1 - x2 - 2 x3 - x5 - 
              x6 ≤ -50, -x1 - x2 - 2 x4 - x5 - 
              x6 ≤ -50, -x1 - x3 - x4 - 2 x5 - 
              x6 ≤ -50, -x2 - x3 - x4 - x5 - 
              2 x6 ≤ -50, -2 x1 - x2 - x3 - 
              x4 ≤ -32, -2 x1 - x2 - x3 - 
              x5 ≤ -32, -2 x1 - x2 - x4 - 
              x5 ≤ -32, -2 x1 - x3 - x4 - 
              x5 ≤ -32, -x1 - 2 x2 - x3 - 
              x4 ≤ -32, -x1 - 2 x2 - x3 - 
              x6 ≤ -32, -x1 - 2 x2 - x4 - 
              x6 ≤ -32, -x1 - x2 - 2 x3 - 
              x5 ≤ -32, -x1 - x2 - 2 x3 - 
              x6 ≤ -32, -x1 - x2 - 2 x4 - 
              x5 ≤ -32, -x1 - x2 - 2 x4 - 
              x6 ≤ -32, -x1 - 2 x3 - x5 - 
              x6 ≤ -32, -x1 - x3 - x4 - 
              2 x5 ≤ -32, -x1 - x3 - 2 x5 - 
              x6 ≤ -32, -x1 - 2 x4 - x5 - 
              x6 ≤ -32, -x1 - x4 - 2 x5 - 
              x6 ≤ -32, -2 x2 - x3 - x4 - 
              x6 ≤ -32, -x2 - 2 x3 - x5 - 
              x6 ≤ -32, -x2 - x3 - x4 - 
              2 x6 ≤ -32, -x2 - x3 - x5 - 
              2 x6 ≤ -32, -x2 - 2 x4 - x5 - 
              x6 ≤ -32, -x2 - x4 - x5 - 
              2 x6 ≤ -32, -x3 - x4 - 2 x5 - 
              x6 ≤ -32, -x3 - x4 - x5 - 
              2 x6 ≤ -32, -x1 - x2 - x3 - 
              x4 ≤ -28, -x1 - x2 - x3 - 
              x5 ≤ -28, -x1 - x2 - x3 - 
              x6 ≤ -28, -x1 - x2 - x4 - 
              x5 ≤ -28, -x1 - x2 - x4 - 
              x6 ≤ -28, -x1 - x3 - x4 - 
              x5 ≤ -28, -x1 - x3 - x5 - 
              x6 ≤ -28, -x1 - x4 - x5 - 
              x6 ≤ -28, -x2 - x3 - x4 - 
              x6 ≤ -28, -x2 - x3 - x5 - 
              x6 ≤ -28, -x2 - x4 - x5 - 
              x6 ≤ -28, -x3 - x4 - x5 - 
              x6 ≤ -28, -x1 - x2 - 
              x4 ≤ -24, -x1 - x3 - 
              x5 ≤ -24, -x2 - x3 - 
              x6 ≤ -24, -x4 - x5 - 
              x6 ≤ -24, -x1 - x2 - 
              x3 ≤ -18, -x1 - x4 - 
              x5 ≤ -18, -x2 - x4 - 
              x6 ≤ -18, -x3 - x5 - 
              x6 ≤ -18, -2 x1 - x2 - 
              x5 ≤ -16, -2 x1 - x3 - 
              x4 ≤ -16, -x1 - 2 x2 - 
              x6 ≤ -16, -x1 - 2 x3 - 
              x6 ≤ -16, -x1 - 2 x4 - 
              x6 ≤ -16, -x1 - 2 x5 - 
              x6 ≤ -16, -2 x2 - x3 - 
              x4 ≤ -16, -x2 - 2 x3 - 
              x5 ≤ -16, -x2 - 2 x4 - 
              x5 ≤ -16, -x2 - x5 - 
              2 x6 ≤ -16, -x3 - x4 - 
              2 x5 ≤ -16, -x3 - x4 - 
              2 x6 ≤ -16, -x1 - x2 - 
              x5 ≤ -14, -x1 - x2 - 
              x6 ≤ -14, -x1 - x3 - 
              x4 ≤ -14, -x1 - x3 - 
              x6 ≤ -14, -x1 - x4 - 
              x6 ≤ -14, -x1 - x5 - 
              x6 ≤ -14, -x2 - x3 - 
              x4 ≤ -14, -x2 - x3 - 
              x5 ≤ -14, -x2 - x4 - 
              x5 ≤ -14, -x2 - x5 - 
              x6 ≤ -14, -x3 - x4 - 
              x5 ≤ -14, -x3 - x4 - 
              x6 ≤ -14, -x1 - 
              x6 ≤ -8, -x2 - 
              x5 ≤ -8, -x3 - 
              x4 ≤ -8, -x1 ≤ 
            0, -x2 ≤ 0, -x3 ≤ 
            0, -x4 ≤ 0, -x5 ≤ 
            0, -x6 ≤ 0, -x1 - x2 - x3 + x6 ≤ 
            0, -x1 - x2 - x3 + x5 ≤ 
            0, -x1 - x2 - x3 + x4 ≤ 
            0, -x1 - x4 - x5 + x6 ≤ 
            0, -x1 + x3 - x4 - x5 ≤ 
            0, -x1 + x2 - x4 - x5 ≤ 
            0, -x2 - x4 + x5 - x6 ≤ 
            0, -x2 + x3 - x4 - x6 ≤ 
            0, -x3 + x4 - x5 - x6 ≤ 0, 
          x2 - x3 - x5 - x6 ≤ 0, x1 - x2 - x4 - x6 ≤ 0, 
          x1 - x3 - x5 - x6 ≤ 
            0, -x1 - x2 + x4 - x5 ≤ 
            6, -x1 - x2 + x4 - x6 ≤ 
            6, -x1 - x2 + x3 - x5 ≤ 
            6, -x1 - x2 + x3 - x6 ≤ 
            6, -x1 - x3 - x4 + x5 ≤ 
            6, -x1 - x3 + x5 - x6 ≤ 
            6, -x1 - x4 + x5 - x6 ≤ 
            6, -x1 + x4 - x5 - x6 ≤ 
            6, -x1 + x3 - x5 - x6 ≤ 
            6, -x1 + x2 - x3 - x4 ≤ 
            6, -x1 + x2 - x3 - x6 ≤ 
            6, -x1 + x2 - x4 - x6 ≤ 
            6, -x2 - x3 - x4 + x6 ≤ 
            6, -x2 - x3 - x5 + x6 ≤ 
            6, -x2 - x4 - x5 + x6 ≤ 
            6, -x2 + x4 - x5 - x6 ≤ 
            6, -x2 + x3 - x5 - x6 ≤ 
            6, -x3 - x4 - x5 + x6 ≤ 
            6, -x3 - x4 + x5 - x6 ≤ 6, 
          x2 - x3 - x4 - x6 ≤ 6, x1 - x2 - x3 - x4 ≤ 6, 
          x1 - x2 - x3 - x5 ≤ 6, x1 - x2 - x4 - x5 ≤ 6, 
          x1 - x3 - x4 - x5 ≤ 6, -x1 - x2 + x4 ≤ 
            12, -x1 - x3 + x5 ≤ 
            12, -x1 + x3 - x5 ≤ 
            12, -x1 + x2 - x4 ≤ 
            12, -x2 - x3 + x6 ≤ 
            12, -x2 + x3 - x6 ≤ 
            12, -x4 - x5 + x6 ≤ 
            12, -x4 + x5 - x6 ≤ 12, 
          x4 - x5 - x6 ≤ 12, x2 - x3 - x6 ≤ 12, 
          x1 - x2 - x4 ≤ 12, 
          x1 - x3 - x5 ≤ 12, -x1 + x6 ≤ 
            14, -x2 + x5 ≤ 14, -x3 + x4 ≤ 14, 
          x3 - x4 ≤ 14, x2 - x5 ≤ 14, 
          x1 - x6 ≤ 14, -x1 + x5 ≤ 
            20, -x1 + x4 ≤ 20, -x1 + x3 ≤ 
            20, -x1 + x2 ≤ 20, -x2 + x6 ≤ 
            20, -x2 + x4 ≤ 20, -x2 + x3 ≤ 
            20, -x3 + x6 ≤ 20, -x3 + x5 ≤ 
            20, -x4 + x6 ≤ 20, -x4 + x5 ≤ 
            20, -x5 + x6 ≤ 20, x5 - x6 ≤ 20, 
          x4 - x5 ≤ 20, x4 - x6 ≤ 20, 
          x3 - x5 ≤ 20, x3 - x6 ≤ 20, 
          x2 - x3 ≤ 20, x2 - x4 ≤ 20, 
          x2 - x6 ≤ 20, x1 - x2 ≤ 20, 
          x1 - x3 ≤ 20, x1 - x4 ≤ 20, 
          x1 - x5 ≤ 20, x6 ≤ 22, x5 ≤ 22, 
          x4 ≤ 22, x3 ≤ 22, x2 ≤ 22, 
          x1 ≤ 22, -x1 + x5 + x6 ≤ 
            32, -x1 + x4 + x6 ≤ 
            32, -x1 + x3 + x6 ≤ 
            32, -x1 + x2 + x6 ≤ 
            32, -x2 + x5 + x6 ≤ 
            32, -x2 + x4 + x5 ≤ 
            32, -x2 + x3 + x5 ≤ 
            32, -x3 + x4 + x6 ≤ 
            32, -x3 + x4 + x5 ≤ 32, 
          x3 - x4 + x6 ≤ 32, x3 - x4 + x5 ≤ 32, 
          x2 - x3 + x4 ≤ 32, x2 - x5 + x6 ≤ 32, 
          x2 + x4 - x5 ≤ 32, x2 + x3 - x4 ≤ 32, 
          x2 + x3 - x5 ≤ 32, x1 - x2 + x5 ≤ 32, 
          x1 - x3 + x4 ≤ 32, x1 + x5 - x6 ≤ 32, 
          x1 + x4 - x6 ≤ 32, x1 + x3 - x4 ≤ 32, 
          x1 + x3 - x6 ≤ 32, x1 + x2 - x5 ≤ 32, 
          x1 + x2 - x6 ≤ 32, x5 + x6 ≤ 34, 
          x4 + x6 ≤ 34, x4 + x5 ≤ 34, 
          x3 + x6 ≤ 34, x3 + x5 ≤ 34, 
          x2 + x6 ≤ 34, x2 + x4 ≤ 34, 
          x2 + x3 ≤ 34, x1 + x5 ≤ 34, 
          x1 + x4 ≤ 34, x1 + x3 ≤ 34, 
          x1 + x2 ≤ 34, x3 + x4 ≤ 36, 
          x2 + x5 ≤ 36, x1 + x6 ≤ 36, 
          x3 + x5 + x6 ≤ 42, x2 + x4 + x6 ≤ 42, 
          x1 + x4 + x5 ≤ 42, x1 + x2 + x3 ≤ 42, 
          x4 + x5 + x6 ≤ 48, x3 + x4 + x6 ≤ 48, 
          x3 + x4 + x5 ≤ 48, x2 + x5 + x6 ≤ 48, 
          x2 + x4 + x5 ≤ 48, x2 + x3 + x6 ≤ 48, 
          x2 + x3 + x5 ≤ 48, x2 + x3 + x4 ≤ 48, 
          x1 + x5 + x6 ≤ 48, x1 + x4 + x6 ≤ 48, 
          x1 + x3 + x6 ≤ 48, x1 + x3 + x5 ≤ 48, 
          x1 + x3 + x4 ≤ 48, x1 + x2 + x6 ≤ 48, 
          x1 + x2 + x5 ≤ 48, 
          x1 + x2 + x4 ≤ 48, -x1 + 2 x5 + x6 ≤ 
            52, -x1 + 2 x4 + x6 ≤ 
            52, -x1 + 2 x3 + x6 ≤ 
            52, -x1 + 2 x2 + x6 ≤ 
            52, -x2 + x5 + 2 x6 ≤ 
            52, -x2 + 2 x4 + x5 ≤ 
            52, -x2 + 2 x3 + x5 ≤ 
            52, -x3 + x4 + 2 x6 ≤ 
            52, -x3 + x4 + 2 x5 ≤ 52, 
          x3 - x4 + 2 x6 ≤ 52, x3 - x4 + 2 x5 ≤ 52, 
          x2 - x5 + 2 x6 ≤ 52, x2 + 2 x4 - x5 ≤ 52, 
          x2 + 2 x3 - x5 ≤ 52, 2 x2 - x3 + x4 ≤ 52, 
          2 x2 + x3 - x4 ≤ 52, x1 + 2 x5 - x6 ≤ 52, 
          x1 + 2 x4 - x6 ≤ 52, x1 + 2 x3 - x6 ≤ 52, 
          x1 + 2 x2 - x6 ≤ 52, 2 x1 - x2 + x5 ≤ 52, 
          2 x1 - x3 + x4 ≤ 52, 2 x1 + x3 - x4 ≤ 52, 
          2 x1 + x2 - x5 ≤ 52, x2 + x3 + x4 + x5 ≤ 54, 
          x1 + x3 + x4 + x6 ≤ 54, 
          x1 + x2 + x5 + x6 ≤ 54, x4 + x5 + 2 x6 ≤ 66, 
          x4 + 2 x5 + x6 ≤ 66, 2 x4 + x5 + x6 ≤ 66, 
          x3 + x4 + 2 x6 ≤ 66, x3 + x4 + 2 x5 ≤ 66, 
          x2 + x5 + 2 x6 ≤ 66, x2 + 2 x4 + x5 ≤ 66, 
          x2 + x3 + 2 x6 ≤ 66, x2 + 2 x3 + x6 ≤ 66, 
          x2 + 2 x3 + x5 ≤ 66, 2 x2 + x3 + x6 ≤ 66, 
          2 x2 + x3 + x4 ≤ 66, x1 + 2 x5 + x6 ≤ 66, 
          x1 + 2 x4 + x6 ≤ 66, x1 + x3 + 2 x5 ≤ 66, 
          x1 + 2 x3 + x6 ≤ 66, x1 + 2 x3 + x5 ≤ 66, 
          x1 + x2 + 2 x4 ≤ 66, x1 + 2 x2 + x6 ≤ 66, 
          x1 + 2 x2 + x4 ≤ 66, 2 x1 + x3 + x5 ≤ 66, 
          2 x1 + x3 + x4 ≤ 66, 2 x1 + x2 + x5 ≤ 66, 
          2 x1 + x2 + x4 ≤ 66, x3 + 2 x4 + x6 ≤ 68, 
          x3 + 2 x4 + x5 ≤ 68, 2 x3 + x4 + x6 ≤ 68, 
          2 x3 + x4 + x5 ≤ 68, x2 + 2 x5 + x6 ≤ 68, 
          x2 + x4 + 2 x5 ≤ 68, x2 + x3 + 2 x5 ≤ 68, 
          x2 + x3 + 2 x4 ≤ 68, x2 + 2 x3 + x4 ≤ 68, 
          2 x2 + x5 + x6 ≤ 68, 2 x2 + x4 + x5 ≤ 68, 
          2 x2 + x3 + x5 ≤ 68, x1 + x5 + 2 x6 ≤ 68, 
          x1 + x4 + 2 x6 ≤ 68, x1 + x3 + 2 x6 ≤ 68, 
          x1 + x3 + 2 x4 ≤ 68, x1 + 2 x3 + x4 ≤ 68, 
          x1 + x2 + 2 x6 ≤ 68, x1 + x2 + 2 x5 ≤ 68, 
          x1 + 2 x2 + x5 ≤ 68, 2 x1 + x5 + x6 ≤ 68, 
          2 x1 + x4 + x6 ≤ 68, 2 x1 + x3 + x6 ≤ 68, 
          2 x1 + x2 + x6 ≤ 68, 
          x3 + 2 x4 + x5 + x6 ≤ 78, 
          x2 + x4 + 2 x5 + x6 ≤ 78, 
          x2 + 2 x3 + x4 + x6 ≤ 78, 
          2 x2 + x3 + x5 + x6 ≤ 78, 
          x1 + x4 + x5 + 2 x6 ≤ 78, 
          x1 + 2 x3 + x4 + x5 ≤ 78, 
          x1 + x2 + x3 + 2 x6 ≤ 78, 
          x1 + x2 + x3 + 2 x5 ≤ 78, 
          x1 + x2 + x3 + 2 x4 ≤ 78, 
          x1 + 2 x2 + x4 + x5 ≤ 78, 
          2 x1 + x3 + x5 + x6 ≤ 78, 
          2 x1 + x2 + x4 + x6 ≤ 78, 
          x1 + x2 + x3 + x4 + x5 + x6 ≤ 78, 
          x2 + x3 + x4 + x5 + 2 x6 ≤ 82, 
          x1 + x3 + x4 + 2 x5 + x6 ≤ 82, 
          x1 + x2 + 2 x4 + x5 + x6 ≤ 82, 
          x1 + x2 + 2 x3 + x5 + x6 ≤ 82, 
          x1 + 2 x2 + x3 + x4 + x6 ≤ 82, 
          2 x1 + x2 + x3 + x4 + x5 ≤ 82, 
          x1 + x2 + x3 + x4 + x5 + 2 x6 ≤ 92, 
          x1 + x2 + x3 + x4 + 2 x5 + x6 ≤ 92, 
          x1 + x2 + x3 + 2 x4 + x5 + x6 ≤ 92, 
          x1 + x2 + 2 x3 + x4 + x5 + x6 ≤ 92, 
          x1 + 2 x2 + x3 + x4 + x5 + x6 ≤ 92, 
          2 x1 + x2 + x3 + x4 + x5 + x6 ≤ 92, 
          x1 + x2 + x3 + x4 + 2 x5 + 2 x6 ≤ 106, 
          x1 + x2 + x3 + 2 x4 + x5 + 2 x6 ≤ 106, 
          x1 + x2 + x3 + 2 x4 + 2 x5 + x6 ≤ 106, 
          x1 + x2 + 2 x3 + x4 + x5 + 2 x6 ≤ 106, 
          x1 + x2 + 2 x3 + x4 + 2 x5 + x6 ≤ 106, 
          x1 + x2 + 2 x3 + 2 x4 + x5 + x6 ≤ 106, 
          x1 + 2 x2 + x3 + x4 + x5 + 2 x6 ≤ 106, 
          x1 + 2 x2 + x3 + x4 + 2 x5 + x6 ≤ 106, 
          x1 + 2 x2 + x3 + 2 x4 + x5 + x6 ≤ 106, 
          x1 + 2 x2 + 2 x3 + x4 + x5 + x6 ≤ 106, 
          2 x1 + x2 + x3 + x4 + x5 + 2 x6 ≤ 106, 
          2 x1 + x2 + x3 + x4 + 2 x5 + x6 ≤ 106, 
          2 x1 + x2 + x3 + 2 x4 + x5 + x6 ≤ 106, 
          2 x1 + x2 + 2 x3 + x4 + x5 + x6 ≤ 106, 
          2 x1 + 2 x2 + x3 + x4 + x5 + x6 ≤ 106, 
          x2 + x3 + x4 + 3 x5 + 2 x6 ≤ 118, 
          x2 + x3 + 3 x4 + x5 + 2 x6 ≤ 118, 
          x2 + 3 x3 + x4 + x5 + 2 x6 ≤ 118, 
          3 x2 + x3 + x4 + x5 + 2 x6 ≤ 118, 
          x1 + x3 + x4 + 2 x5 + 3 x6 ≤ 118, 
          x1 + x3 + 3 x4 + 2 x5 + x6 ≤ 118, 
          x1 + 3 x3 + x4 + 2 x5 + x6 ≤ 118, 
          x1 + x2 + 2 x4 + x5 + 3 x6 ≤ 118, 
          x1 + x2 + 2 x4 + 3 x5 + x6 ≤ 118, 
          x1 + x2 + 2 x3 + x5 + 3 x6 ≤ 118, 
          x1 + x2 + 2 x3 + 3 x5 + x6 ≤ 118, 
          x1 + 2 x2 + x3 + x4 + 3 x6 ≤ 118, 
          x1 + 2 x2 + x3 + 3 x4 + x6 ≤ 118, 
          x1 + 2 x2 + 3 x3 + x4 + x6 ≤ 118, 
          x1 + 3 x2 + 2 x4 + x5 + x6 ≤ 118, 
          x1 + 3 x2 + 2 x3 + x5 + x6 ≤ 118, 
          2 x1 + x2 + x3 + x4 + 3 x5 ≤ 118, 
          2 x1 + x2 + x3 + 3 x4 + x5 ≤ 118, 
          2 x1 + x2 + 3 x3 + x4 + x5 ≤ 118, 
          2 x1 + 3 x2 + x3 + x4 + x5 ≤ 118, 
          3 x1 + x3 + x4 + 2 x5 + x6 ≤ 118, 
          3 x1 + x2 + 2 x4 + x5 + x6 ≤ 118, 
          3 x1 + x2 + 2 x3 + x5 + x6 ≤ 118, 
          3 x1 + 2 x2 + x3 + x4 + x6 ≤ 118, 
          x1 + x2 + x3 + 2 x4 + 2 x5 + 2 x6 ≤ 120, 
          x1 + 2 x2 + 2 x3 + x4 + x5 + 2 x6 ≤ 120, 
          2 x1 + x2 + 2 x3 + x4 + 2 x5 + x6 ≤ 120, 
          2 x1 + 2 x2 + x3 + 2 x4 + x5 + x6 ≤ 120, 
          x1 + x2 + x3 + x4 + 2 x5 + 3 x6 ≤ 124, 
          x1 + x2 + x3 + x4 + 3 x5 + 2 x6 ≤ 124, 
          x1 + x2 + x3 + 2 x4 + x5 + 3 x6 ≤ 124, 
          x1 + x2 + x3 + 2 x4 + 3 x5 + x6 ≤ 124, 
          x1 + x2 + x3 + 3 x4 + x5 + 2 x6 ≤ 124, 
          x1 + x2 + x3 + 3 x4 + 2 x5 + x6 ≤ 124, 
          x1 + x2 + 2 x3 + x4 + x5 + 3 x6 ≤ 124, 
          x1 + x2 + 2 x3 + x4 + 3 x5 + x6 ≤ 124, 
          x1 + x2 + 3 x3 + x4 + x5 + 2 x6 ≤ 124, 
          x1 + x2 + 3 x3 + x4 + 2 x5 + x6 ≤ 124, 
          x1 + 2 x2 + x3 + x4 + x5 + 3 x6 ≤ 124, 
          x1 + 2 x2 + x3 + 3 x4 + x5 + x6 ≤ 124, 
          x1 + 2 x2 + 3 x3 + x4 + x5 + x6 ≤ 124, 
          x1 + 3 x2 + x3 + x4 + x5 + 2 x6 ≤ 124, 
          x1 + 3 x2 + x3 + 2 x4 + x5 + x6 ≤ 124, 
          x1 + 3 x2 + 2 x3 + x4 + x5 + x6 ≤ 124, 
          2 x1 + x2 + x3 + x4 + 3 x5 + x6 ≤ 124, 
          2 x1 + x2 + x3 + 3 x4 + x5 + x6 ≤ 124, 
          2 x1 + x2 + 3 x3 + x4 + x5 + x6 ≤ 124, 
          2 x1 + 3 x2 + x3 + x4 + x5 + x6 ≤ 124, 
          3 x1 + x2 + x3 + x4 + 2 x5 + x6 ≤ 124, 
          3 x1 + x2 + x3 + 2 x4 + x5 + x6 ≤ 124, 
          3 x1 + x2 + 2 x3 + x4 + x5 + x6 ≤ 124, 
          3 x1 + 2 x2 + x3 + x4 + x5 + x6 ≤ 124, 
          x1 + x2 + 2 x3 + 2 x4 + x5 + 3 x6 ≤ 134, 
          x1 + x2 + 2 x3 + 2 x4 + 3 x5 + x6 ≤ 134, 
          x1 + 2 x2 + x3 + x4 + 2 x5 + 3 x6 ≤ 134, 
          x1 + 2 x2 + x3 + 3 x4 + 2 x5 + x6 ≤ 134, 
          x1 + 2 x2 + 3 x3 + x4 + 2 x5 + x6 ≤ 134, 
          x1 + 3 x2 + 2 x3 + 2 x4 + x5 + x6 ≤ 134, 
          2 x1 + x2 + x3 + x4 + 3 x5 + 2 x6 ≤ 134, 
          2 x1 + x2 + x3 + 3 x4 + x5 + 2 x6 ≤ 134, 
          2 x1 + x2 + 3 x3 + x4 + x5 + 2 x6 ≤ 134, 
          2 x1 + 3 x2 + x3 + x4 + x5 + 2 x6 ≤ 134, 
          3 x1 + x2 + 2 x3 + 2 x4 + x5 + x6 ≤ 134, 
          3 x1 + 2 x2 + x3 + x4 + 2 x5 + x6 ≤ 134, 
          x1 + x2 + x3 + 2 x4 + 2 x5 + 4 x6 ≤ 156, 
          x1 + x2 + x3 + 2 x4 + 4 x5 + 2 x6 ≤ 156, 
          x1 + x2 + x3 + 4 x4 + 2 x5 + 2 x6 ≤ 156, 
          x1 + 2 x2 + 2 x3 + x4 + x5 + 4 x6 ≤ 156, 
          x1 + 2 x2 + 4 x3 + x4 + x5 + 2 x6 ≤ 156, 
          x1 + 4 x2 + 2 x3 + x4 + x5 + 2 x6 ≤ 156, 
          2 x1 + x2 + 2 x3 + x4 + 4 x5 + x6 ≤ 156, 
          2 x1 + x2 + 4 x3 + x4 + 2 x5 + x6 ≤ 156, 
          2 x1 + 2 x2 + x3 + 4 x4 + x5 + x6 ≤ 156, 
          2 x1 + 4 x2 + x3 + 2 x4 + x5 + x6 ≤ 156, 
          4 x1 + x2 + 2 x3 + x4 + 2 x5 + x6 ≤ 156, 
          4 x1 + 2 x2 + x3 + 2 x4 + x5 + x6 ≤ 156, 
          x1 + x2 + 2 x3 + 4 x4 + x5 + 3 x6 ≤ 170, 
          x1 + x2 + 2 x3 + 4 x4 + 3 x5 + x6 ≤ 170, 
          x1 + x2 + 4 x3 + 2 x4 + x5 + 3 x6 ≤ 170, 
          x1 + x2 + 4 x3 + 2 x4 + 3 x5 + x6 ≤ 170, 
          x1 + 2 x2 + x3 + x4 + 4 x5 + 3 x6 ≤ 170, 
          x1 + 2 x2 + x3 + 3 x4 + 4 x5 + x6 ≤ 170, 
          x1 + 2 x2 + 3 x3 + x4 + 4 x5 + x6 ≤ 170, 
          x1 + 3 x2 + 2 x3 + 4 x4 + x5 + x6 ≤ 170, 
          x1 + 3 x2 + 4 x3 + 2 x4 + x5 + x6 ≤ 170, 
          x1 + 4 x2 + x3 + x4 + 2 x5 + 3 x6 ≤ 170, 
          x1 + 4 x2 + x3 + 3 x4 + 2 x5 + x6 ≤ 170, 
          x1 + 4 x2 + 3 x3 + x4 + 2 x5 + x6 ≤ 170, 
          2 x1 + x2 + x3 + x4 + 3 x5 + 4 x6 ≤ 170, 
          2 x1 + x2 + x3 + 3 x4 + x5 + 4 x6 ≤ 170, 
          2 x1 + x2 + 3 x3 + x4 + x5 + 4 x6 ≤ 170, 
          2 x1 + 3 x2 + x3 + x4 + x5 + 4 x6 ≤ 170, 
          3 x1 + x2 + 2 x3 + 4 x4 + x5 + x6 ≤ 170, 
          3 x1 + x2 + 4 x3 + 2 x4 + x5 + x6 ≤ 170, 
          3 x1 + 2 x2 + x3 + x4 + 4 x5 + x6 ≤ 170, 
          3 x1 + 4 x2 + x3 + x4 + 2 x5 + x6 ≤ 170, 
          4 x1 + x2 + x3 + x4 + 3 x5 + 2 x6 ≤ 170, 
          4 x1 + x2 + x3 + 3 x4 + x5 + 2 x6 ≤ 170, 
          4 x1 + x2 + 3 x3 + x4 + x5 + 2 x6 ≤ 170, 
          4 x1 + 3 x2 + x3 + x4 + x5 + 2 x6 ≤ 
            170};

The cells below define the equations used in section 5.1.

General[spell1::off];
VarNames[n_] := 
        Flatten[Table[{z[j] -> 
                ToExpression[StringJoin["z", ToString[j]]], 
              G[j] -> 
                ToExpression[StringJoin["G", ToString[j]]]}, {j, 1, 
              n}]];
VarNames2 := 
        Flatten[Table[
            s[i, j] -> 
              ToExpression[
                StringJoin["s", ToString[i], ToString[j]]], {i, 1, 
              3}, {j, i + 1, 4}]];
s[i_, i_] := 0;
s[i_, j_] := s[j, i] /; j < i;
S[i_, i_] :=  0;
S[i_, j_]  := 1/s[i, j] - 1/1;
Z[i_, j_] := 0 /; i == j;
Z[i_, j_] := 1 /; i ≠ j;
Sigma[i_, j_, n_]  :=  
        2*(G[i] + G[j]) S[i, j] + 
          Plus @@ Table[
              G[h]*Z[i, h]*Z[j, h]*(S[i, h] + S[j, h]), {h, 1, 
                n}];
P[i_, j_, n_]  := 
        s[i, j]*Sigma[i, j, n] + 
          Plus @@ Table[
              G[h]*Z[i, h]*
                Z[j, h]*(s[i, h] - s[j, h])*(S[i, h] - S[j, h]), {h, 
                1, n}];
CCeqs[
          n_]  :=  Table[P[i, j, n], {j, 2, n}, {i, 1, j - 1}] // 
              Flatten // Factor // 
          Numerator;
ACEquations  = CCeqs[4] /. VarNames[4] /. 
          VarNames2;
dz1  =  (1/s12 - 1)*(1/s34 - 1) - (1/s14 - 
                    1) (1/s23 - 1) // Factor // 
          Numerator;
dz2  =  (1/s12 - 1)*(1/s34 - 1) - (1/s13 - 
                    1) (1/s24 - 1) // Factor // 
          Numerator;
dz3  =  (1/s13 - 1)*(1/s24 - 1) - (1/s14 - 
                    1) (1/s23 - 1) // Factor // 
          Numerator;
NineEquations  =  
        ACEquations~Join~{dz1, dz2, dz3};
GetExponents[mon_, vars_]  :=  
      Map[Exponent[mon, #] &, vars]
GetAllExponents[poly_, 
        vars_] := 
Table[
          GetExponents[poly[[i]], vars], {i, 1, Length[poly]}] // 
        Union
vars4  =  {s12, s13, s14, s23, s24, 
          s34};
NineEquationsExponents  =  
        Map[GetAllExponents[#, vars4] &, NineEquations];

The following functions are used to extract the nontrivial facets of the Minkowski sum polytope.

Subst[v_]  :=  {x1 -> v[[1]],  x2 -> v[[2]],  
          x3 -> v[[3]], x4 -> v[[4]], 
          x5 -> v[[5]], 
          x6 -> v[[6]]};
lhs[LessEqual[a_, b_]]  :=  a;
rhs[LessEqual[a_, b_]]  :=  b;
FindMax[linform_, vertlist_]  :=  
        Max[ Map[linform /. Subst[#] &, vertlist]];

    
GetMaximalExponents[ineq_, 
          vertlist_] := 
With[{linform = lhs[ineq], 
            maxval  =  FindMax[lhs[ineq], vertlist]},  
          Select[vertlist, ((linform /. Subst[#])  == 
                  maxval) &]];

  (*returns 1 if the inequality determines a reduced polynomial with more than one term *) 
Nontrivial[ineq_, vertlist_] :=  
        If[Length[GetMaximalExponents[ineq, vertlist]] > 1, 1, 
          0];

  
AllNontrivial[ineq_, vllist_]  := 
If[
          Min[Map[Nontrivial[ineq, #] &, vllist]] > 0, True, 
          False];

There are 35 nontrivial facets of the Minkowski sum polytope.  Here are their defining inequalities, sorted according to the value of a linear function on their inward normal vectors a.

nontrivialfacets  =  
        Select[v1t9bIeq, 
          AllNontrivial[#, 
              NineEquationsExponents] &];
Length[%]

GetAlpha[
        LessEqual[a_, b_]] :=  -Coefficient[
          a, {x1, x2, x3, x4, x5, 
            x6}]

nontrivialfacets  =  
        Sort[nontrivialfacets, ({1, 1, 1, 1, 1, 1} . 
                  GetAlpha[#1] > {1, 1, 1, 1, 1, 1} . 
                  GetAlpha[#2]) &];

{Table[i, {i, 1, Length[nontrivialfacets]}], nontrivialfacets} // 
        Transpose // MatrixForm

Besides the facets of the Minkowski sum polytope, we need to detect nontrivial faces of lower dimension.  We will use these functions to work our way down from the facets.

FindFacet[ineq_, vlist_] := 
        Select[vlist, ((lhs[ineq] /. Subst[#]) === 
                rhs[ineq]) &];
FacetsIncident[indlist_, ineqlist_, vlist_]  :=  
        With[{f = 
              Map[FindFacet[#, vlist] &, 
                Part[ineqlist, indlist]]}, 
If[
            Length[Intersection @@ f] > 0, True, 
            False]];
AddLEInequalities[indlist_, ineqlist_] := 
        Module[{ins, rights,  lefts}, 
ins  =  
            Part[ineqlist, indlist]; 
rights  =  
            Map[rhs, ins]; 
lefts  =  
            Map[lhs, ins]; 
LessEqual[Plus @@ lefts, 
            Plus @@ rights]];
NontrivialFace[indlist_, ineqlist_, explist_]  :=  
        With[{in = 
              AddLEInequalities[indlist, 
                ineqlist]}, 
AllNontrivial[in, 
            explist]];

The symmetry of our equations under the action S_4, the permutation group, can be exploited to reduce the amount of analysis required.  The following functions are for that purpose.

Off[General::spell1];
sigma4  =  Permutations[{1, 2, 3, 4}];
k[i_, i_] := 0;
k[i_, j_] := k[j, i] /; j < i;
kvars  =  {k[1, 2], k[1, 3], k[1, 4], k[2, 3], k[2, 4], 
          k[3, 4]};
kineqs  =  
        nontrivialfacets /. {x1 -> k[1, 2], x2 -> k[1, 3], 
            x3 -> k[1, 4], x4 -> k[2, 3], x5 -> k[2, 4], 
            x6 -> k[3, 4]};
Permute[perm_, k[i_, j_]] :=  
        k[perm[[i]], perm[[j]]];
Permute[perm_, LessEqual[a_, b_]] :=  
        With[{coeffs =  Coefficient[a, kvars]}, 
LessEqual[
            coeffs . Map[Permute[perm, #] &, kvars], b]];
PermutedIndex[perm_, ineqlist_, ind_]  :=  
        With[{pineq  =  
              Permute[perm, 
                ineqlist[[ind]]]}, 
Position[
              ineqlist, pineq][[1, 1]]
];
PermutedIndexList[perm_, ineqlist_, indlist_]  :=  
        Sort[Map[PermutedIndex[perm, ineqlist, #] &, 
            indlist]];
IndexListOrbit[pgroup_, ineqlist_, indlist_] :=  
        Union[Map[PermutedIndexList[#, ineqlist, indlist] &, 
            pgroup]];
MinimalOrbitRep[pgroup_, ineqlist_, indlist_] :=  
        First[IndexListOrbit[pgroup, ineqlist, 
            indlist]];
ExtendIndexList[ineqlist_, {}]  :=  
        Table[{i}, {i, 1, Length[ineqlist]}];
ExtendIndexList[ineqlist_, indlist_]  :=  
        Table[indlist~Join~{i}, {i, Last[indlist] + 1, 
            Length[ineqlist]}];
ExtendedMinimalOrbitReps[pgroup_, ineqlist_, indlist_]  :=  
        Module[{i, ext, minrep, oreps = {}}, 
ext  =  
            ExtendIndexList[ineqlist, indlist]; 
For[i = 1, 
            i ≤ Length[ext], i++, 
minrep = 
              MinimalOrbitRep[pgroup, ineqlist, 
                ext[[i]]]; 
AppendTo[oreps, 
              minrep];
]; 
oreps // 
            Union];
ExtendedMinimalOrbitRepsList[pgroup_, ineqlist_, 
          indlistlist_]  :=  
Map[
              ExtendedMinimalOrbitReps[pgroup, ineqlist, #] &, indlistlist] // 
            Flatten[#, 1] & // Union;

Using the above functions, we first find the nontrivial facet representatives under the action of the permutation group.  These are listed in table 1 of the paper and numbered 1-9.

faces1   =  
      ExtendedMinimalOrbitRepsList[sigma4, kineqs, {{}}]

nontrivialfacetreps = 
      Sort[Map[nontrivialfacets[[#]] &, 
          Flatten[faces1]], ({1, 1, 1, 1, 1, 1} . 
                GetAlpha[#1] > {1, 1, 1, 1, 1, 1} . 
                GetAlpha[#2]) &]

Now we work our way down in dimension, finding lower-dimension faces of the Minkowski sum polytope that correspond to nontrivial reduced equations.

faces2   =  
        ExtendedMinimalOrbitRepsList[sigma4, kineqs, 
          faces1];
faces2  =  
      Select[faces2, (NontrivialFace[#, nontrivialfacets, 
                NineEquationsExponents]  && 
              FacetsIncident[#, nontrivialfacets, v1t9bH]) &]

faces3   =  
        ExtendedMinimalOrbitRepsList[sigma4, kineqs, 
          faces2];
faces3  =  
      Select[faces3, (NontrivialFace[#, nontrivialfacets, 
                NineEquationsExponents]  && 
              FacetsIncident[#, nontrivialfacets, v1t9bH]) &]

faces4   =  
        ExtendedMinimalOrbitRepsList[sigma4, kineqs, 
          faces3];
faces4  =  
      Select[faces4, (NontrivialFace[#, nontrivialfacets, 
                NineEquationsExponents]  && 
              FacetsIncident[#, nontrivialfacets, v1t9bH]) &]

We only need to test faces which have at least one inward normal a in the half-space 
-a1 - ... - a6 >= 0.  In the list of nontrivial facets, only facets 22 - 35 have such a normal.  Here are all the indices of the nontrivial faces which intersect at least one of these facets and therefore have an inward normal in the half-space.  We will have to examine their reduced equations below.

nontrivialfacetrepsIndex = {{22}, {25}, {29}, {33}, {8, 25}, {12, 
            29}, {12, 33}, {16, 35}, {22, 33}, {12, 16, 35}};

In addition to identifying the nontrivial faces of the polytopes, we must also compute the corresponding reduced equations.  The following functions perform these computations.

ReducePoly[p_, ineq_]  := 
        Module[{q, linform, maxval, explist}, 
q = 
            Expand[p]; 
linform  =  
            lhs[ineq]; 
explist  =  
            GetAllExponents[q, vars4]; 
maxval  =  
            FindMax[linform, explist]; 
Select[
            q, ((linform /. Subst[GetExponents[#, vars4]]) == 
                  maxval) &]
];
ReducedSystem[LessEqual[lhs_, rhs_], eqlist_]  :=  
        Map[ReducePoly[#, LessEqual[lhs, rhs]] &, eqlist];
ReducedSystem[indlist_, ineqlist_, eqlist_]  :=  
        ReducedSystem[AddLEInequalities[indlist, ineqlist], 
          eqlist];
RemoveFactor[poly_, var_]  := 
        Module[{i, pow, p}, 
p  =  
            Expand[poly]; 
pow =  
            Min[Table[
                Exponent[p[[i]], var], {i, 1, 
                  Length[p]}]]; 
If[
            pow > 0 && pow < Infinity, (poly/var^pow) // Factor, 
            poly]];
CleanUp[poly_, vars_]  :=  
Fold[RemoveFactor, 
          poly, vars ];

Any reduced system is homogeneous with respect to some weight vector.  In searching for nonzero solutions, we can use this homogeneity to require an additional condition, that the polynomial nzeq defined below is zero:

nzeq  =  s12*s13*s14*s23*s24*s34 - 1;

The ReductionElimination function eliminates the s_{ij} variables from a given set of reduced equations, to obtain polynomial conditions in the G_i that would have to hold for a corresponding solution curve to exist.

ReductionElimination[indlist_, ineqlist_, eqlist_] := 
      Module[{redeqs, gb}, 
redeqs =  
          ReducedSystem[indlist, ineqlist, eqlist] // 
            Factor; 
redeqs  = 
          Map[CleanUp[#, vars4] &, redeqs]; 
gb = 
          GroebnerBasis[redeqs, Union[{G1, G2, G3, G4}, vars4], 
            MonomialOrder -> 
              DegreeReverseLexicographic]; 
gb = 
          GroebnerBasis[gb~Join~{nzeq}, Union[{G1, G2, G3, G4}, vars4], 
            MonomialOrder -> 
              DegreeReverseLexicographic]; 
gb = 
          GroebnerBasis[gb, {G1, G2, G3, G4}, vars4]; 
gb = 
          GroebnerBasis[gb~Join~{G1*G2*G3*G4*w - 1}, {G1, G2, G3, G4}, {w}]; 
        Factor[GroebnerBasis[
            gb~Join~{(G1 + G2 + G3 + G4)*w - 1}, {G1, G2, G3, 
              G4}, {w}]]
]

The following computations may take some time.  For convenience we include the results so the reader can skip the evaluation.
	The results show two sets of problematic vorticities.  The first, when polynomials of the form G1 G2-G3 G4 = 0 are satisfied, can be dealt with by considering higher order Puiseux expansions.  These calculations follow below.  In the second case, we are restricted to vorticities of the form k(-1,1,1,1), for which we can compute an explicit Groebner basis.

Map[
      ReductionElimination[#, nontrivialfacets, 
          NineEquations] &, {{22}, {25}, {29}, {33}}]

The computation below shows that for lower-dimensional faces, problems can only occur if a subsum of two or three of the G_i vanishes.

Map[
      ReductionElimination[#, nontrivialfacets, NineEquations] &, {{8, 
          25}, {12, 29}, {12, 33}, {16, 35}, {22, 33}, {12, 16, 
          35}}]

The remainder of this section is devoted to showing that extending the Puiseux series further results in a contradiction.  First we analyze the reduced system of facet 22 (which is called number 6 in table 1 and in section 5.1.1 of the paper)  to find the leading terms in the Puiseux expansion.  Fortunately, there is a unique solution up to the expected homogeneity.

red  =  
        ReducedSystem[{22}, nontrivialfacets, NineEquations] // 
          Factor;
red  = Map[CleanUp[#, vars4] &, red];
Factor[red]

SimplifiedRed22 = 
      Complement[
        Factor[red /. {s24 -> 1 - s13, 
              s23 -> 
                1 - s14}], {0}]

Factor[
      GroebnerBasis[
        Join[SimplifiedRed22, {G1 G2 - G3 G4, 
            G1*G2*G3*G4*s34*w - 1}], {s12, s13, s14, s34, G1, G2, G3, 
          G4}, {w}]]

Note that if some pairs of vorticities add to zero, such as G1+G3=0, then there are no nonzero solutions to the reduced equations for nonzero vorticities.  This justifies the substitutions used below in solving the reduced system.

Solve[-G1 + G1 s14 + G3 s14 == 0, 
      s14]
Solve[-G1 + G1 s13 + G4 s13 == 0, 
      s13]

Numerator[
      Factor[red /. {s24 -> 1 - s13, s23 -> 1 - s14, 
              s34 -> -s12} /. {s14 -> (G1/(G1 + G3)), 
            s13 -> (G1/(G1 + G4))}]]

Here are the values of the leading coefficients.  s34 =  s12 but the value is arbitrary and could be normalized.

vars4 /. {s24 -> 1 - s13, s23 -> 1 - s14, 
          s34 -> -s12} /. {s14 -> (G1/(G1 + G3)), 
        s13 -> (G1/(G1 + G4))}

Next, we make a change of variable so that the leading terms of the Puiseux series are constants.  In an abuse of notation, our new variables have the same name as the old (s_{ij}).  As a check on our calculations, we verify that upon substituting our new variables into the equations the constant terms are zero.

NewNineEquations = 
      Numerator[
        Factor[(NineEquations/t) /. {s12 -> s12 t^(- 1), 
                s34 -> s34 t^(- 1)} /. {G4 -> 
                G1*G2/G3}]]

red0 = 
        Factor[Simplify[
            NewNineEquations /. {t -> 0}]];
red0  = 
      Factor[Map[CleanUp[#, vars4] &, 
          red0]]

sub1 = {s24 -> 1 - (G1/(G1 + G4)), 
        s23 -> 1 - (G1/(G1 + G3)), s34 -> -s12, 
        s14 -> (G1/(G1 + G3)), 
        s13 -> (G1/(G1 + G4))}
Factor[red0 /. sub1 /. G4 -> G1*G2/G3]

Next we show that the reduced system has full rank.  It follows that the Puiseux series in these variables is actually a power series.

grad[f_]  :=  {D[f, s13], D[f, s14], D[f, s23], D[f, s24], 
        D[f, s34]}
DF0  =  Map[grad, red0] /. sub1 /. {G4 -> G1*G2/G3} // 
          Factor;
DF0 // MatrixForm

Union[Factor[Minors[DF0, 5]]][[60]]

Finally we show that the series cannot be continued consistently to the first order in t:

NewNineEquationsO1 = 
      NewNineEquations /. {s12 -> s12 + t*u12, s13 -> s13 + t*u13, 
          s14 -> s14 + t*u14, s23 -> s23 + t*u23, 
          s24 -> s24 + t*u24, 
          s34 -> 
            s34 + t*u34}

Ft1  = 
      Numerator[ Coefficient[NewNineEquationsO1, t] /. 
              sub1 /. {G4 -> G1*G2/G3} // 
          Factor]

Factor[
      GroebnerBasis[
        Union[Ft1, {s12*G1*G2*G3*w - 1}], {w, s12, u12, u13, u14, u23, u24, 
          u34, G1, G2, G3}]]

Strictly Planar Relative Equilibria with zero total circulation

Finally, we verify the calculations from section 5.2, which show that there are at most 14 strictly planar relative equilibria when the total circulation (G1+G2+G3+G4) is zero.  As usual, we begin by setting up the equations.  s34 is normalized to be 1.

s[i_, i_] := 0;
s[i_, j_] := 
        s[j, i] /; j < i;

S[i_] := Sum[G[j] s[i, j], {j, 1, 4}] /. VarNames[4] /. 
        VarNames2

SEquations1 = 
      Table[S[i] - S[i + 1], {i, 1, 3}] /. {s34 -> 
            1}

SLagrange = {s12^(-1) + s34^(-1), 
          s13^(-1) + s24^(-1), 
          s14^(-1) + s23^(-1)};
SEquations2 = 
      Numerator[
        Factor[Table[
              SLagrange[[Mod[i, 3] + 1]] - 
                SLagrange[[Mod[i + 1, 3] + 1]], {i, 1, 
                3}] /. {s34 -> 1}]]

Next we eliminate three of the variables.

SolvingFors12s13s23 = Solve[
          SEquations1 == {0, 0, 0}, {s12, s13, 
            s23}][[1]]

SEquations2AfterSubstitution = 
      Numerator[Factor[SEquations2 /. SolvingFors12s13s23]]

There are only two variables left, s14 and s24. Computing resultants yields a complicated degree 7 polynomial in s14.

PairwiseResultants = {Factor[
          Resultant[SEquations2AfterSubstitution[[1]], 
            SEquations2AfterSubstitution[[2]], s24]], 
        Factor[Resultant[SEquations2AfterSubstitution[[2]], 
            SEquations2AfterSubstitution[[3]], 
            s24]]}

p7 = 
        PairwiseResultants[[2]]/(-16 (G1^2) (G2^5) G3 (G1 + 
                  G2 - G3 - G4) G4);

p7Coefficients = 
        Table[Coefficient[p7, s14, i], {i, 0, 
            7}];

Factor[
      GroebnerBasis[
        Union[p7Coefficients, {G1  +  G2  +  G3  +  G4,  
            G1 G2 G3 G4  - 1}], {G1, G2, G3, G4}, 
        MonomialOrder -> DegreeReverseLexicographic]]

SpecialCaseEquations = 
      Union[SEquations1, SEquations2] /. {G1 -> 1, G2 -> -1, 
          G3 -> 1, G4 -> -1, s34 -> 1}

GroebnerBasis[
      SpecialCaseEquations, {s12, s13, s14, s23, s24}]

Adding in our normalization s34=1, we find explicitly the solutions to the special case:

Solve[{-2 + 8 s24 - 11 (s24^2) + 6 (s24^3) - (s24^4), -7 + 
            s23 + 16 s24 - 11 (s24^2) + 2 (s24^3), -7 + s14 + 16 s24 - 
            11 (s24^2) + 2 (s24^3), -16 + s13 + 33 s24 - 22 (s24^2) + 
            4 (s24^3), -1 + s12, 1 - s34} == {0, 0, 0, 0, 0, 
          0}, {s12, s13, s14, s23, s24, s34}]