<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:
        # Get the dimensions of the grid
        m, n = len(grid), len(grid[0])
        
        # Initialize prefix sum arrays for rows, columns, diagonals, and anti-diagonals
        rows = [[0] * (n + 1) for _ in range(m)]
        cols = [[0] * n for _ in range(m + 1)]
        diag = [[0] * (n + 1) for _ in range(m + 1)]
        anti = [[0] * (n + 1) for _ in range(m + 1)]
        
        # Calculate prefix sums for rows, columns, diagonals, and anti-diagonals
        for i in range(m):
            for j in range(n):
                rows[i][j + 1] = grid[i][j] + rows[i][j]
                cols[i + 1][j] = grid[i][j] + cols[i][j]
                diag[i + 1][j + 1] = grid[i][j] + diag[i][j]
                anti[i + 1][j] = grid[i][j] + anti[i][j + 1]
        
        # Initialize the maximum size of the magic square found
        ans = 1
        
        # Iterate over each possible top-left corner of a square
        for i in range(m):
            for j in range(n):
                # Try to form a square of size k+1 starting from (i, j)
                for k in range(1, min(i, j) + 1):
                    ii, jj = i - k, j - k
                    # Calculate the sum of the main diagonal
                    val = diag[i + 1][j + 1] - diag[ii][jj]
                    # Check if the anti-diagonal has the same sum
                    match = (val == anti[i + 1][jj] - anti[ii][j + 1])
                    
                    # Check if all row sums are equal to the diagonal sum
                    for r in range(ii, i + 1):
                        match &= (val == rows[r][j + 1] - rows[r][jj])
                    
                    # Check if all column sums are equal to the diagonal sum
                    for c in range(jj, j + 1):
                        match &= (val == cols[i + 1][c] - cols[ii][c])
                    
                    # If all conditions are met, update the maximum size
                    if match:
                        ans = max(ans, k + 1)
        
        # Return the size of the largest magic square found
        return ans
933