<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, n = len(board), len(board[0])

        rows = [[] for _ in range(m)]

        cols = [[] for _ in range(n)]


        idxes = defaultdict(list)

        for i in range(m):
            for j in range(n):
                num = -board[i][j]
                rows[i].append(num)
                cols[j].append(num)
                idxes[num].append((i, j))
                if len(rows[i]) > 3:
                    rows[i].sort()
                    rem = rows[i].pop()
                    ri, rj = idxes[rem].pop()
                    cols[rj].remove(rem)
                if len(cols[j]) > 3:
                    cols[j].sort()
                    rem = cols[j].pop()
                    for ri, rj in idxes[rem]:
                        if rj == j: break
                    idxes[rem].remove((ri, rj))
                    rows[ri].remove(rem)
        
        arr = []

        for num in sorted(idxes.keys()):
            for coor in idxes[num]:
                arr.append((-num, coor))

        def dfs(i, r1=-1, c1=-1, r2=-1, c2=-1):
            if i == len(arr):
                return -inf
            val, coor = arr[i]
            r, c = coor
            if r2 != -1 and r not in (r1, r2) and c not in (c1, c2):
                return val
            res = dfs(i+1, r1, c1, r2, c2)
            if r1 == -1:
                res = max(res, val + dfs(i+1, r, c, r2, c2))
            elif r2 == -1 and r != r1 and c != c1:
                res = max(res, val + dfs(i+1, r1, c1, r, c))
            return res
        
        return dfs(0)
63778