<p>A <code>k x k</code> <strong>magic square</strong> is a <code>k x k</code> grid filled with integers such that every row sum, every column sum, and both diagonal sums are <strong>all equal</strong>. The integers in the magic square <strong>do not have to be distinct</strong>. Every <code>1 x 1</code> grid is trivially a <strong>magic square</strong>.</p>

<p>Given an <code>m x n</code> integer <code>grid</code>, return <em>the <strong>size</strong> (i.e., the side length </em><code>k</code><em>) of the <strong>largest magic square</strong> that can be found within this grid</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/05/29/magicsquare-grid.jpg" style="width: 413px; height: 335px;" />
<pre>
<strong>Input:</strong> grid = [[7,1,4,5,6],[2,5,1,6,4],[1,5,4,3,2],[1,2,7,3,4]]
<strong>Output:</strong> 3
<strong>Explanation:</strong> The largest magic square has a size of 3.
Every row sum, column sum, and diagonal sum of this magic square is equal to 12.
- Row sums: 5+1+6 = 5+4+3 = 2+7+3 = 12
- Column sums: 5+5+2 = 1+4+7 = 6+3+3 = 12
- Diagonal sums: 5+4+3 = 6+4+2 = 12
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/05/29/magicsquare2-grid.jpg" style="width: 333px; height: 255px;" />
<pre>
<strong>Input:</strong> grid = [[5,1,3,1],[9,3,3,1],[1,3,3,8]]
<strong>Output:</strong> 2
</pre>

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

<ul>
	<li><code>m == grid.length</code></li>
	<li><code>n == grid[i].length</code></li>
	<li><code>1 &lt;= m, n &lt;= 50</code></li>
	<li><code>1 &lt;= grid[i][j] &lt;= 10<sup>6</sup></code></li>
</ul>

class Solution:
    def largestMagicSquare(self, grid: List[List[int]]) -> int:
        m, n = len(grid), len(grid[0])
        rowPreSum = [[0] * (n + 1) for _ in range(m)]
        colPreSum = [[0] * n for _ in range(m + 1)]

        for i in range(m):
            for j in range(n):
                rowPreSum[i][j + 1] = rowPreSum[i][j] + grid[i][j]
                
        for j in range(n):
            for i in range(m):
                colPreSum[i + 1][j] = colPreSum[i][j] + grid[i][j]

        def check(row: int, col: int, sz: int) -> bool:
            target = rowPreSum[row][col + sz] - rowPreSum[row][col]

            for i in range(row + 1, row + sz):
                if rowPreSum[i][col + sz] - rowPreSum[i][col] != target:
                    return False

            for j in range(col, col + sz):
                if colPreSum[row + sz][j] - colPreSum[row][j] != target:
                    return False

            diagonalSum = reverseDiagonalSum = 0
            for k in range(sz):
                diagonalSum += grid[row + k][col + k]
                reverseDiagonalSum += grid[row + k][col + sz - 1 - k]

            return diagonalSum == reverseDiagonalSum == target

        ans = 0
        for i in range(m):
            for j in range(n):
                for sz in range(1, min(m - i + 1, n - j + 1)):
                    if check(i, j, sz):
                        ans = max(ans, sz)

        return ans
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