<p>You are given a 2D matrix <code>grid</code> consisting of positive integers.</p>

<p>You have to select <em>one or more</em> cells from the matrix such that the following conditions are satisfied:</p>

<ul>
	<li>No two selected cells are in the <strong>same</strong> row of the matrix.</li>
	<li>The values in the set of selected cells are <strong>unique</strong>.</li>
</ul>

<p>Your score will be the <strong>sum</strong> of the values of the selected cells.</p>

<p>Return the <strong>maximum</strong> score you can achieve.</p>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">grid = [[1,2,3],[4,3,2],[1,1,1]]</span></p>

<p><strong>Output:</strong> <span class="example-io">8</span></p>

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

<p><img alt="" src="https://assets.leetcode.com/uploads/2024/07/29/grid1drawio.png" /></p>

<p>We can select the cells with values 1, 3, and 4 that are colored above.</p>
</div>

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

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

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

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

<p><img alt="" src="https://assets.leetcode.com/uploads/2024/07/29/grid8_8drawio.png" style="width: 170px; height: 114px;" /></p>

<p>We can select the cells with values 7 and 8 that are colored above.</p>
</div>

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

<ul>
	<li><code>1 &lt;= grid.length, grid[i].length &lt;= 10</code></li>
	<li><code>1 &lt;= grid[i][j] &lt;= 100</code></li>
</ul>

class Solution:
    def maxScore(self, grid: List[List[int]]) -> int:
        # for row in grid:
        #     row.sort()
        
        m = len(grid)
        n = len(grid[0])

        distinct = set()

        for r in range(m):
            for c in range(n):
                distinct.add(grid[r][c])

        @cache
        def backtrack(count, cur, c):
            if count >= min(len(distinct), m):
                return cur
            
            # if (count, cur, c) in d:
            #     return d[(count, cur, c)]

            res = 0
            for r in range(m):
                if r not in picked:
                    picked.add(r)
                    for c in range(n):
                        if grid[r][c] not in used:
                            used.add(grid[r][c])
                            res = max(res, backtrack(count + 1, cur + grid[r][c], c))
                            used.remove(grid[r][c])
                        else:
                            res = max(res, backtrack(count + 1, cur, c))
                    picked.remove(r)
            
            # d[(count, cur, c)] = res
            return res

        picked = set()
        used = set()

        d = defaultdict(int)

        res = backtrack(0, 0, 0)
        
        return res
        


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