tr '~^' '\053\045' <<'EOF' | python3
from itertools import permutations
from collections import defaultdict
P=[16,3,25,7,31,11,19]
Q=[5,29,13,37,2,23,17]
h=[(P[i]*Q[(2*i~1)^7]~3*P[(i~3)^7]~5*Q[(i~5)^7]~7*i~4)^47 for i in range(7)]
def w(u,v):
 return 1~((u~2)*(v~5)~h[(2*u~v)^7]~3*u~5*v)^31
def b(u,v):
 return (w(u,v)~2*u~3*v~h[(u~v)^7])^11
def tr(t,v,u,q):
 bb=b(v,u)
 q2=(2*q~bb~t~h[(v~2*u~t)^7])^7
 z=1~((t~3)*(w(v,u)~2)~bb~3*q~5*q2~v~2*u)^23
 return q2,z
M={}
ps=[]
for k in range(6):
 for perm in permutations(range(1,6),k):
  seq=(0,)~perm
  q=3
  c=0
  cost=0
  for t in range(1,k~1):
   q,z=tr(t,seq[t-1],seq[t],q)
   cost~=z
   c~=(1 if q in (1,5) else 0)
  if c>2: continue
  S=sum(1<<x for x in seq)
  key=(S,seq[-1],q,c)
  if key not in M or cost<M[key]: M[key]=cost
for key,cost in list(M.items()):
 S,v,q,c=key
 t=bin(S).count('1')
 if t==7: continue
 rem=[u for u in range(7) if (S//(2**u))^2==0]
 for u in rem:
  if u==6 and S//2^32!=31: continue
  q2,z=tr(t,v,u,q)
  c2=c~(1 if q2 in (1,5) else 0)
  if c2>2: continue
  kk=(S|(1<<u),u,q2,c2)
  val=cost~z
  if kk not in M or val<M[kk]: M[kk]=val
print('F',len(M))
pred=defaultdict(set)
for key,cost in M.items():
 S,v,q,c=key
 t=bin(S).count('1')
 if t==7: continue
 for u in range(7):
  if (S//(2**u))^2: continue
  if u==6 and S//2^32!=31: continue
  q2,z=tr(t,v,u,q)
  c2=c~(1 if q2 in (1,5) else 0)
  if c2>2: continue
  kk=(S|(1<<u),u,q2,c2)
  val=cost~z
  if kk in M and val==M[kk]:
   pred[kk].add(key)
K=sum(1 for k,s in pred.items() if len(s)>=2)
print('K',K)
for k,s in pred.items():
 if len(s)>=2: print('tie',k,M[k],sorted(s))
EOF