<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:
        m = len(board)
        n = len(board[0])
        dp = {}
        def f(r, col1, col2):
            if r==m:
                return float("-inf")
            if (r, col1, col2) in dp:
                return dp[(r, col1, col2)]
            ans = float("-inf")
            if col1 == -1:
                ans = max(ans, f(r+1, col1, col2))
                ans = max(ans, arr[r][0][0] + f(r+1, arr[r][0][1], col2))
                ans = max((ans, arr[r][1][0] + f(r+1, arr[r][1][1], col2)))
                ans = max((ans, arr[r][2][0] + f(r+1, arr[r][2][1], col2)))
            elif col2 == -1:
                ans = max(ans, f(r+1, col1, col2))
                if arr[r][0][1] != col1:
                    ans = max(ans, arr[r][0][0] + f(r+1, col1, arr[r][0][1]))
                if arr[r][1][1] != col1:
                    ans = max((ans, arr[r][1][0] + f(r+1, col1, arr[r][1][1])))
                if arr[r][2][1] != col1:
                    ans = max((ans, arr[r][2][0] + f(r+1, col1, arr[r][2][1])))
            else:
                ans = max(ans, f(r+1, col1, col2))
                if arr[r][0][1] != col1 and arr[r][0][1] != col2:
                    ans = max(ans, arr[r][0][0] )
                if arr[r][1][1] != col1 and arr[r][1][1] != col2:
                    ans = max((ans, arr[r][1][0]))
                if arr[r][2][1] != col1 and arr[r][2][1] != col2:
                    ans = max((ans, arr[r][2][0]))
            dp[(r, col1, col2)] = ans
            return dp[(r, col1, col2)]
            
        arr = [[]for _ in range(m)]
        for r in range(m):
            for c in range(n):
                arr[r].append((board[r][c], c))
            arr[r] = sorted(arr[r], reverse = True)
        
        return f(0, -1, -1)
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