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 = '# https://zxi.mytechroad.com/blog/searching/leetcode-996-number-of-squareful-arrays/\n# g[i][j]: if i and j are squareful\n# dp[s][i]: # of ways to reach state s (binary mask of nodes visited) that ends with node i\n# dp[s | (1 << j)][j] += dp[s][i] if g[i][j]\n# 先对数组排序, 然后计算出所有的 g[i][j]\n# for i in range(n) 初始化 dp[(1 << i)][i] = 1, 表示只有一个数字 i 的情况, 每个都是一种\n# 如果有多个重复的数字, 只用第一个数字作为开头\n# ans = sum(dp[(1 << n) - 1][i])\n# O(n^2*2^n) time complexity, O(2^n) space complexity\nimport math\nclass Solution:\n    def numSquarefulPerms(self, A: List[int]) -> int:\n        n = len(A)\n        A = sorted(A)\n        g = [[False] * n for _ in range(n)]\n        dp = [[0] * n for _ in range(1 << n)]\n        \n        for i in range(n):\n            for j in range(n):\n                if i == j:\n                    continue\n                if int((A[i] + A[j]) ** 0.5 + 0.5) ** 2 == A[i] + A[j]:\n                    g[i][j] = True\n\n        for i in range(n):\n            if i > 0 and A[i] == A[i - 1]:  # For the same numbers, only the first one can be the starting point\n                continue\n            dp[1 << i][i] = 1\n\n        for s in range(1 << n):\n            for i in range(n):\n                if dp[s][i] <= 0:  # 如果 dp[s][i] <= 0, 说明没有以 i 结尾的情况, 跳过\n                    continue\n                for j in range(n):  # 尝试让任意一个数字连接在最后面\n                    if not g[i][j] or (s & (1 << j)):  # 如果 i, j 不是 squareful 或 j 已经使用过, 跳过\n                        continue\n                    if j > 0 and not (s & (1 << (j - 1))) and A[j] == A[j - 1]:  # 出现重复的只使用第一个\n                        continue\n                    dp[s | (1 << j)][j] += dp[s][i]\n        \n        res = 0\n        for i in range(n):\n            res += dp[(1 << n) - 1][i]\n        return res\n'
_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: 7396