<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 = len(grid)
    n = len(grid[0])
    # prefixRow[i][j] := the sum of the first j numbers in the i-th row
    prefixRow = [[0] * (n + 1) for _ in range(m)]
    # prefixCol[i][j] := the sum of the first j numbers in the i-th column
    prefixCol = [[0] * (m + 1) for _ in range(n)]

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

    def isMagicSquare(i: int, j: int, k: int) -> bool:
      """Returns True if grid[i..i + k)[j..j + k) is a magic square."""
      diag, antiDiag = 0, 0
      for d in range(k):
        diag += grid[i + d][j + d]
        antiDiag += grid[i + d][j + k - 1 - d]
      if diag != antiDiag:
        return False
      for d in range(k):
        if self._getSum(prefixRow, i + d, j, j + k - 1) != diag:
          return False
        if self._getSum(prefixCol, j + d, i, i + k - 1) != diag:
          return False
      return True

    def containsMagicSquare(k: int) -> bool:
      """Returns True if the grid contains any magic square of size k x k."""
      for i in range(m - k + 1):
        for j in range(n - k + 1):
          if isMagicSquare(i, j, k):
            return True
      return False

    for k in range(min(m, n), 1, -1):
      if containsMagicSquare(k):
        return k

    return 1

  def _getSum(self, prefix: list[list[int]], i: int, l: int, r: int) -> int:
    """Returns sum(grid[i][l..r]) or sum(grid[l..r][i])."""
    return prefix[i][r + 1] - prefix[i][l]
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