python3 <<'PY'
N={i:[] for i in range(1,9)}
for e in (12,23,34,45,56,67,78,18,25,36,47):
 a,b=divmod(e,10)
 N[a].append(b)
 N[b].append(a)
for v in N: N[v].sort()
adj=lambda a,b: b in N[a]
mod13=lambda x: pow(x,1,13)
par=lambda x: pow(x,1,2)
def lr(T): return (T[1],T[2],T[0])
def rr(T): return (T[2],T[0],T[1])
def phi(T,b,c):
 e=T[b]
 U=[x for x in N[e] if x not in T]
 if not U: return lr(T) if c==1 else T
 x=min(U) if c==0 else max(U)
 L=list(T)
 L[b]=x
 return tuple(L)
def psi(T,b,c):
 j=2-b
 e=T[j]
 U=[x for x in N[e] if x not in T]
 if not U: return rr(T) if c==0 else T
 x=max(U) if c==0 else min(U)
 L=list(T)
 L[j]=x
 return tuple(L)
def P(T): return [sum(1 for x in T if adj(v,x)) for v in range(1,9)]
def sig(T,Q): return mod13(sum([T[0],2*T[1],3*T[2],Q[T[0]-1],2*Q[T[1]-1],3*Q[T[2]-1]]))
T0s={0:(1,4,7),1:(2,5,8),2:(3,6,1)}
wa={0:2,1:3,2:2}
wbc={0:3,1:2}
wd={0:3,1:1}
we={0:1,1:1}
num=0
den=0
for A in range(3):
 T0=T0s[A]
 for B in range(2):
  for C in range(2):
   F=1 if (8 in T0 or par(sum([T0[0],T0[2]]))==0) else 0
   G0=1 if sum(1 for x in T0 if x in (2,4,6,8))>=2 else 0
   X=phi(T0,B,C)
   Y=psi(T0,B,C)
   S=0 if X==T0 else 1
   SX=sum(set(X))
   SY=sum(set(Y))
   J=1 if (Y!=X and SX==SY) else 0
   if F==1 and S==0:
    T1=T0
    src='T0'
   elif J==1 and par(sum([A,B,C]))==0:
    T1=Y
    src='Y'
   else:
    T1=X
    src='X'
   Tpre=T1
   P0=P(T1)
   r0=sig(T1,P0)
   Z0=1 if (par(sum([P0[1],P0[4],B]))==1 and (adj(T1[0],T0[1]) or adj(T1[2],T0[0]))) else 0
   T1r=T1
   if Z0==1:
    if src=='X':
     T1r=Y if par(SX)==par(SY) else T0
    elif src=='Y':
     T1r=lr(T0)
    else:
     T1r=X
   P1=P(T1r)
   if Z0==1:
    m=len(set(Tpre).symmetric_difference(set(T1r)))
    r1=mod13(sum([sig(T1r,P1),r0,m]))
   else:
    P1=P0
    r1=r0
   for D in range(2):
    for E in range(2):
     T2=T1r
     Q=0
     R=r1
     if D==1:
      if par(sum([R,Z0]))==0:
       T2=T0
       Q=1
       if adj(X[1],T0[0]):
        R=r0
        Q=2
      else:
       T2=rr(T1r)
       Q=3
       R=mod13(sum([R,T2[0],-T1r[0]]))
     if E==1:
      if Q==2:
       R=mod13(sum([R,P0[2],P0[5],A]))
      elif Q==3:
       cnt=sum(1 for x in set(X) if adj(x,T0[1]))
       R=mod13(sum([R,cnt]))
      else:
       R=mod13(sum([R,T2[1] if G0==1 else T2[2]]))
     Zstar=1 if (par(sum([P0[0],P0[7],C]))==0 and (adj(T2[1],T0[2]) or adj(T2[0],Tpre[0]))) else 0
     if Zstar!=Z0:
      R=mod13(sum([r0,sum(set(T2)), X[0] if Zstar==1 else Y[2]]))
      T2=lr(T2) if Zstar==1 else rr(T2)
     Zdag=1 if (par(sum([P1[3],P0[6],A,B]))==1 and (adj(T2[2],T0[0]) or adj(Tpre[0],T2[1]))) else 0
     if Zdag==1:
      if Zstar==1:
       T2=(T2[1],T2[0],T2[2])
      else:
       T2=rr(T2)
      R=mod13(sum([r1,sum(set(T2)),P0[T2[0]-1], Y[1] if C==1 else X[1]]))
     Z1=1 if (adj(T2[0],T0[2]) or (P1[6]<P0[6] and par(R)==0) or (Zstar==1 and Z0==0 and len(set(T2).intersection(set(T0)))==2) or (Zdag==1 and R in (0,5,9))) else 0
     if Z1==1:
      R=mod13(sum([R,sum(set(T0)), Y[0] if Z0==1 else X[2], Zstar, 2*Zdag]))
     L=1 if (R in (0,6,7) and Z1==par(sum([A,B,S,J,E])) and par(sum([R,A,C,Z0,Zstar,Zdag]))==D) else 0
     H=1 if (L==1 and F==1 and S==1 and C==0 and Zstar!=Z0 and Z1==1 and R in (0,6,7)) else 0
     w=wa[A]*wbc[B]*wbc[C]*wd[D]*we[E]
     if L==1: den=den+w
     if H==1: num=num+w
     if L==1 or H==1:
      print(A,B,C,D,E,'L',L,'H',H,'R',R,'Z1',Z1,'Z0',Z0,'Zs',Zstar,'Zd',Zdag,'T0',T0,'Tpre',Tpre,'T1r',T1r,'T2',T2,'r0',r0,'r1',r1,'P0',P0,'P1',P1,'X',X,'Y',Y,'Q',Q,'w',w)
print('TOTALS',num,den)
PY