Given an array A of non-negative integers, the array is squareful if for every pair of adjacent elements, their sum is a perfect square.
Return the number of permutations of A that are squareful.  Two permutations A1 and A2 differ if and only if there is some index i such that A1[i] != A2[i].
 
Example 1:
Input: [1,17,8]
Output: 2
Explanation: 
[1,8,17] and [17,8,1] are the valid permutations.

Example 2:
Input: [2,2,2]
Output: 1

 
Note:

1 <= A.length <= 12
0 <= A[i] <= 1e9

```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 = 'from typing import List\nfrom collections import defaultdict\n\nclass Solution:\n    def helper(self, last_num, used, partners, index):\n        if index == len(used):\n            return 1\n        total = 0\n        # print(partners, partners[last_num])\n        for partner, partner_list in partners[last_num].items():\n            for partner_index in partner_list:\n                if not used[partner_index]:\n                    used[partner_index] = True\n                    total += self.helper(partner, used, partners, index+1)\n                    used[partner_index] = False\n                    break\n        return total\n\n    def numSquarefulPerms(self, A: List[int]) -> int:\n        squares = set()\n        i = 0\n        partners = defaultdict(lambda: defaultdict(set))\n        while i * i <= 200000000:\n            squares.add(i*i)\n            i += 1\n        for i in range(len(A)):\n            for j in range(i+1, len(A), 1):\n                if A[i] + A[j] in squares:\n                    partners[A[i]][A[j]].add(j)\n                    partners[A[j]][A[i]].add(i)\n        unique_nums = dict()\n        for i, num in enumerate(A):\n            unique_nums[num] = i\n        used = [False for _ in A]\n        total = 0\n        for num, i in unique_nums.items():\n            used[i] = True\n            total += self.helper(num, used, partners, 1)\n            used[i] = False\n        return total'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'numSquarefulPerms'
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: 985920