A game on an undirected graph is played by two players, Mouse and Cat, who alternate turns.
The graph is given as follows: graph[a] is a list of all nodes b such that ab is an edge of the graph.
Mouse starts at node 1 and goes first, Cat starts at node 2 and goes second, and there is a Hole at node 0.
During each player's turn, they must travel along one edge of the graph that meets where they are.  For example, if the Mouse is at node 1, it must travel to any node in graph[1].
Additionally, it is not allowed for the Cat to travel to the Hole (node 0.)
Then, the game can end in 3 ways:

If ever the Cat occupies the same node as the Mouse, the Cat wins.
If ever the Mouse reaches the Hole, the Mouse wins.
If ever a position is repeated (ie. the players are in the same position as a previous turn, and it is the same player's turn to move), the game is a draw.

Given a graph, and assuming both players play optimally, return 1 if the game is won by Mouse, 2 if the game is won by Cat, and 0 if the game is a draw.
 



Example 1:
Input: [[2,5],[3],[0,4,5],[1,4,5],[2,3],[0,2,3]]
Output: 0
Explanation:
4---3---1
|   |
2---5
 \ /
  0

 
Note:

3 <= graph.length <= 50
It is guaranteed that graph[1] is non-empty.
It is guaranteed that graph[2] contains a non-zero element.

```python

USER_FILENAME = "<user_code>"
import sys, json, builtins
import math, heapq, bisect, itertools, functools, collections, statistics, fractions, decimal, operator, array, random, re, string, os, pathlib
try:
    import resource as _res
except Exception:
    _res = None
from collections import Counter, defaultdict, deque, OrderedDict
from functools import lru_cache, reduce
from itertools import accumulate, permutations, combinations, product
from math import gcd, ceil, floor, sqrt
from decimal import Decimal
try:
    from typing import *
except Exception:
    pass
try:
    sys.setrecursionlimit(1_000_000)
except Exception:
    pass
try:
    import site as _site
    try:
        _sys_sites = _site.getsitepackages() if hasattr(_site, "getsitepackages") else []
        for _p in _sys_sites:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
    try:
        _usp = _site.getusersitepackages()
        if isinstance(_usp, str):
            _usp = [_usp]
        for _p in _usp:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
except Exception:
    pass

def _disabled_input(*args, **kwargs):
    raise RuntimeError("input() is disabled in callable mode")
builtins.input = _disabled_input

ns = {"__name__": "__judge__"}  # prevent if __name__ == "__main__" blocks
for _k, _v in list(globals().items()):
    if not _k.startswith("_"):
        ns[_k] = _v
_CODE = "class Solution:\n    # O(n^4) time, O(n^3) space\n    def catMouseGame(self, graph: List[List[int]]) -> int:\n        n = len(graph)\n        edges = defaultdict(list)\n        for i, nbrs in enumerate(graph):\n            for nbr in nbrs:\n                edges[i].append(nbr)\n                edges[nbr].append(i)\n        \n        # cache = {}\n        @lru_cache(None)\n        def can_win(cat_pos, mouse_pos, remaining_moves, is_cats_turn):\n            # state = cat_pos, mouse_pos, is_cats_turn, remaining_moves\n            # if state in cache:\n            #     return cache[state]\n            if mouse_pos == 0:  # mouse wins\n                if is_cats_turn:\n                    # cache[state] = -1\n                    return -1\n                # cache[state] = 1\n                return 1\n            if cat_pos == mouse_pos:  # cat wins\n                if is_cats_turn:\n                    # cache[state] = 1\n                    return 1\n                # cache[state] = -1\n                return -1\n            if remaining_moves == 0:  # draw\n                # cache[state] = 0\n                return 0\n            \n            opponents_worst_case = 1\n            if is_cats_turn:\n                for nbr in edges[cat_pos]:\n                    if nbr == 0:\n                        continue\n                    opponents_worst_case = min(\n                        opponents_worst_case, can_win(nbr, mouse_pos, remaining_moves - 1, not is_cats_turn)\n                    )\n            else:\n                for nbr in edges[mouse_pos]:\n                    opponents_worst_case = min(\n                        opponents_worst_case, can_win(cat_pos, nbr, remaining_moves - 1, not is_cats_turn)\n                    )\n            \n            # cache[state] = -opponents_worst_case\n            return -opponents_worst_case\n        \n        mouse_outcome = can_win(2, 1, 2 * n, False)\n        if mouse_outcome == 1:\n            return 1\n        if mouse_outcome == -1:\n            return 2\n        return 0\n    \n#     def catMouseGame(self, graph: List[List[int]]) -> int:\n#         n = len(graph)\n#         edges = defaultdict(list)\n#         for i, nbrs in enumerate(graph):\n#             for nbr in nbrs:\n#                 edges[i].append(nbr)\n#                 edges[nbr].append(i)\n        \n#         q = [(1, 2)]\n#         seen = set()\n#         seen.add((1, 2))\n#         step = 0\n#         is_mouses_turn = True\n#         while q:\n#             if step > 2 * n:\n#                 break\n#             # last round is cat's turn\n#             if is_mouses_turn and any(cat_pos == mouse_pos for mouse_pos, cat_pos in q):\n#                 return 2\n#             # last round is mouse's turn\n#             if not is_mouses_turn and any(mouse_pos == 0 for mouse_pos, _ in q):\n#                 return 1\n            \n#             q2 = []\n#             for mouse_pos, cat_pos in q:\n#                 if is_mouses_turn:\n#                     for nbr in edges[mouse_pos]:\n#                         next_state = nbr, cat_pos\n#                         if next_state in seen:\n#                             continue\n#                         q2.append(next_state)\n#                         seen.add(next_state)\n#                 else:\n#                     for nbr in edges[cat_pos]:\n#                         if nbr == 0:\n#                             continue\n#                         next_state = mouse_pos, nbr\n#                         if next_state in seen:\n#                             continue\n#                         q2.append(next_state)\n#                         seen.add(next_state)\n#             q = q2\n#             step += 1\n#             is_mouses_turn = not is_mouses_turn\n            \n#         return 0\n            \n        \n    def catMouseGame(self, graph: List[List[int]]) -> int:\n        @lru_cache(None)\n        def search(t, x, y):\n            if t == len(graph) * 2: return 0\n            if x == y: return 2\n            if x == 0:return 1\n            if (t%2==0):# mouse's turn. Mouse will win if the mouse can find any winable node for the next step. If all the next step is winable for cats, then mouse lose.\n                if any(search(t+1, x_nxt, y)==1 for x_nxt in graph[x]):return 1\n                if all(search(t+1, x_nxt, y)==2 for x_nxt in graph[x]):return 2\n                return 0\n            else:# cat's turn\n                if any(search(t+1, x, y_nxt)==2 for y_nxt in graph[y] if y_nxt!=0):return 2\n                if all(search(t+1, x, y_nxt)==1 for y_nxt in graph[y] if y_nxt!=0):return 1\n                return 0\n        return search(0, 1, 2)"
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'catMouseGame'
if _fn_name:
    if "Solution" in ns:
        try:
            _obj = ns["Solution"]()
            if hasattr(_obj, _fn_name):
                _target = getattr(_obj, _fn_name)
        except Exception:
            _target = None
    if _target is None and _fn_name in ns and callable(ns[_fn_name]):
        _target = ns[_fn_name]
# Fallback: if there's exactly one user-defined function, use it
if _target is None:
    _cands = [v for k, v in ns.items() if callable(v) and getattr(v, "__module__", "") in ("__judge__", "__main__")]
    _user_funcs = [c for c in _cands if getattr(c, "__name__", "").startswith("_") is False]
    if len(_user_funcs) == 1:
        _target = _user_funcs[0]
if _target is None:
    raise RuntimeError(f"Could not resolve callable '{_fn_name}'")

_raw = sys.stdin.read().strip()
if _raw:
    _parsed = json.loads(_raw)
    if isinstance(_parsed, dict):
        _kwargs = _parsed.get("kwargs", {})
        _args = _parsed.get("args", [])
    else:
        _args, _kwargs = _parsed, {}
else:
    _args, _kwargs = [], {}
if not isinstance(_args, (list, tuple)):
    _args = [_args]
_res = _target(*_args, **_kwargs)
try:
    sys.stdout.write(repr(_res) + "\n")
except Exception:
    pass

```

Measured memory_bytes: 28352