<p>You are given an integer array <code>matchsticks</code> where <code>matchsticks[i]</code> is the length of the <code>i<sup>th</sup></code> matchstick. You want to use <strong>all the matchsticks</strong> to make one square. You <strong>should not break</strong> any stick, but you can link them up, and each matchstick must be used <strong>exactly one time</strong>.</p>

<p>Return <code>true</code> if you can make this square and <code>false</code> otherwise.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/04/09/matchsticks1-grid.jpg" style="width: 253px; height: 253px;" />
<pre>
<strong>Input:</strong> matchsticks = [1,1,2,2,2]
<strong>Output:</strong> true
<strong>Explanation:</strong> You can form a square with length 2, one side of the square came two sticks with length 1.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> matchsticks = [3,3,3,3,4]
<strong>Output:</strong> false
<strong>Explanation:</strong> You cannot find a way to form a square with all the matchsticks.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= matchsticks.length &lt;= 15</code></li>
	<li><code>1 &lt;= matchsticks[i] &lt;= 10<sup>8</sup></code></li>
</ul>

class Solution:
    def __init__(self):
        self.matchsticks = []
        self.found = False
        self.target = 0
    
    @functools.lru_cache(None)
    def f(self, count: int, curr: int, used: int) -> None:
        if self.found:
            return
        
        if count >= 3:
            self.found = True
            return

        for i in range(len(self.matchsticks)):
            mask = 1 << i
            if mask & used or curr + self.matchsticks[i] > self.target:
                continue
            nused = used
            nused = nused | mask

            if curr + self.matchsticks[i] < self.target:
                self.f(count, curr + self.matchsticks[i], nused)
            elif curr + self.matchsticks[i] == self.target:
                self.f(count + 1, 0, nused)

    def makesquare(self, matchsticks: List[int]) -> bool:
        if sum(matchsticks) % 4 != 0:
            return False

        self.matchsticks = sorted(matchsticks, reverse=True)
        self.target = sum(matchsticks) // 4
        self.f(0,0, 0)
        return self.found
        
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