<p>You are given a <code>m x n</code> 2D array <code>board</code> representing a chessboard, where <code>board[i][j]</code> represents the <strong>value</strong> of the cell <code>(i, j)</code>.</p>

<p>Rooks in the <strong>same</strong> row or column <strong>attack</strong> each other. You need to place <em>three</em> rooks on the chessboard such that the rooks <strong>do not</strong> <strong>attack</strong> each other.</p>

<p>Return the <strong>maximum</strong> sum of the cell <strong>values</strong> on which the rooks are placed.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">board = </span>[[-3,1,1,1],[-3,1,-3,1],[-3,2,1,1]]</p>

<p><strong>Output:</strong> 4</p>

<p><strong>Explanation:</strong></p>

<p><img alt="" src="https://assets.leetcode.com/uploads/2024/08/08/rooks2.png" style="width: 294px; height: 450px;" /></p>

<p>We can place the rooks in the cells <code>(0, 2)</code>, <code>(1, 3)</code>, and <code>(2, 1)</code> for a sum of <code>1 + 1 + 2 = 4</code>.</p>
</div>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">board = [[1,2,3],[4,5,6],[7,8,9]]</span></p>

<p><strong>Output:</strong> <span class="example-io">15</span></p>

<p><strong>Explanation:</strong></p>

<p>We can place the rooks in the cells <code>(0, 0)</code>, <code>(1, 1)</code>, and <code>(2, 2)</code> for a sum of <code>1 + 5 + 9 = 15</code>.</p>
</div>

<p><strong class="example">Example 3:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">board = [[1,1,1],[1,1,1],[1,1,1]]</span></p>

<p><strong>Output:</strong> <span class="example-io">3</span></p>

<p><strong>Explanation:</strong></p>

<p>We can place the rooks in the cells <code>(0, 2)</code>, <code>(1, 1)</code>, and <code>(2, 0)</code> for a sum of <code>1 + 1 + 1 = 3</code>.</p>
</div>

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

<ul>
	<li><code>3 &lt;= m == board.length &lt;= 500</code></li>
	<li><code>3 &lt;= n == board[i].length &lt;= 500</code></li>
	<li><code>-10<sup>9</sup> &lt;= board[i][j] &lt;= 10<sup>9</sup></code></li>
</ul>

class Solution:
    def maximumValueSum(self, board: List[List[int]]) -> int:
        m = len(board)
        n = len(board[0])
        board_size = n * m

        options = set()
        rows = [set() for _ in range(n)]
        columns = [set() for _ in range(m)]
        for j in range(m):
            for i in range(n):
                value = board[j][i]
                option = (i, j, value)
                options.add(option)
                rows[i].add(option)
                columns[j].add(option)

        options_sorted = sorted(options, key=itemgetter(2), reverse=True)

        max_sum = None

        i = j = k = 0
        for i in range(board_size):
            max_possible_sum = sum(option[2] for option in options_sorted[i : i + 3])
            if max_sum == max_possible_sum:
                break
            left = options_sorted[i]
            for j in range(i + 1, board_size):
                if max_sum == max_possible_sum:
                    break
                middle = options_sorted[j]
                if (
                    max_sum is not None
                    and j < board_size - 1
                    and max_sum > left[2] + middle[2] + options_sorted[j + 1][2]
                ):
                    break
                if middle in rows[left[0]]:
                    continue
                if middle in columns[left[1]]:
                    continue
                for k in range(j + 1, board_size):
                    if max_sum == max_possible_sum:
                        break
                    right = options_sorted[k]
                    if max_sum is not None and max_sum > left[2] + middle[2] + right[2]:
                        break
                    if right in rows[left[0]]:
                        continue
                    if right in columns[left[1]]:
                        continue
                    if right in rows[middle[0]]:
                        continue
                    if right in columns[middle[1]]:
                        continue
                    values_sum = left[2] + middle[2] + right[2]
                    if max_sum is None or values_sum > max_sum:
                        max_sum = values_sum

        return max_sum

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