<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;= 100</code></li>
	<li><code>3 &lt;= n == board[i].length &lt;= 100</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:
        num_rows, num_cols = len(board), len(board[0])
        top_vals_by_row = []
        for cells in board:
            top_inds = sorted(range(num_cols), key=lambda i: cells[i], reverse=True)[:3]
            top_vals_by_row.append(
                [(cells[i], i) for i in top_inds]
            )

        import functools

        INFTY = -int(1E13)

        @functools.cache
        def find_max_val_from_suffix(row: int, col1, col2):
            return max(
                val
                for i in range(row + 1, num_rows)
                for val, col in top_vals_by_row[i]
                if col != col1 and col != col2
            )

        @functools.cache
        def find_max_val(row: int, prev_chosen_col: int | None):
            if row == num_rows:
                return INFTY
            result = find_max_val(row + 1, prev_chosen_col)
            if prev_chosen_col is None:
                for curr_val, curr_col in top_vals_by_row[row]:
                    result = max(curr_val + find_max_val(row + 1, curr_col), result)
                return result
            assert prev_chosen_col is not None
            if row + 1 == num_rows:
                return result
            for curr_val, curr_col in top_vals_by_row[row]:
                if curr_col == prev_chosen_col:
                    continue
                result = max(
                    curr_val + find_max_val_from_suffix(row, prev_chosen_col, curr_col),
                    result,
                )
            return result

        return find_max_val(0, None)
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