#!/usr/bin/env python3
import time,math,functools
N=112;F={}
for s in range(2,2*N+1):
 F[s]=[(u,s//u) for u in range(1,min(N,math.isqrt(s))+1) if s%u==0 and s//u<=N]
t=time.time();vis=set();cnt=0;depth=0
def search(T):
 global cnt,depth
 if T in vis:return False
 vis.add(T);cnt+=1;depth=max(depth,len(T))
 S={a+b for a in T for b in T};P={a*b for a in T for b in T}
 if S<=P:print('FOUND',T);return True
 D=S-P; ifopts=[]
 for s in D:
  opts=[(u,v) for u,v in F[s] if not ((113-u in T and u not in T) or (113-v in T and v not in T) or (u+v==113))]
  if not opts:return False
  ifopts.append((len(opts),s,opts))
 _,s,opts=min(ifopts)
 for u,v in opts:
  if search(tuple(sorted(set(T)|{u,v}))):return True
 return False
print('result',search((1,2,3,5,7)),'cnt',cnt,'depth',depth,'secs',round(time.time()-t,2))