<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:
        row, col = len(grid), len(grid[0])

        val_state = []
        for r in range(row):
            for c, val in enumerate(grid[r]):
                val_state.append([val, r])
        val_state.sort(key=lambda x: x[0], reverse=True)
        groups = []
        for val, r in val_state:
            if not groups or groups[-1][0][0] != val:
                groups.append([[val, r]])
            else:
                groups[-1].append([val, r])
        dp = collections.defaultdict(int)

        def helper(group_index, state):
            if group_index == len(groups):
                return 0
            key = (group_index, state)
            if key in dp:
                return dp[key]
            for current_val, current_r in groups[group_index]:
                # case1, not to chose
                option1 = helper(group_index + 1, state)

                #case2, chose the current one
                option2 = 0
                if state & (1 << current_r) == 0:
                    option2 = helper(group_index + 1, state | (1 << current_r)) + current_val
                dp[key] = max(dp[key], max(option1, option2))
            return dp[key]
        return helper(0, 0)
59960