You are given a 2D matrix grid of size n x n. Initially, all cells of the grid are colored white. In one operation, you can select any cell of indices (i, j), and color black all the cells of the jth column starting from the top row down to the ith row.
The grid score is the sum of all grid[i][j] such that cell (i, j) is white and it has a horizontally adjacent black cell.
Return the maximum score that can be achieved after some number of operations.
\n\n\n
Example 1:
\n\nInput: grid = [[0,0,0,0,0],[0,0,3,0,0],[0,1,0,0,0],[5,0,0,3,0],[0,0,0,0,2]]
\n\nOutput: 11
\n\nExplanation:
\n![\"\"](\"https://assets.leetcode.com/uploads/2024/05/11/one.png\")
\nIn the first operation, we color all cells in column 1 down to row 3, and in the second operation, we color all cells in column 4 down to the last row. The score of the resulting grid is grid[3][0] + grid[1][2] + grid[3][3] which is equal to 11.
Example 2:
\n\nInput: grid = [[10,9,0,0,15],[7,1,0,8,0],[5,20,0,11,0],[0,0,0,1,2],[8,12,1,10,3]]
\n\nOutput: 94
\n\nExplanation:
\n![\"\"](\"https://assets.leetcode.com/uploads/2024/05/11/two-1.png\")
\nWe perform operations on 1, 2, and 3 down to rows 1, 4, and 0, respectively. The score of the resulting grid is grid[0][0] + grid[1][0] + grid[2][1] + grid[4][1] + grid[1][3] + grid[2][3] + grid[3][3] + grid[4][3] + grid[0][4] which is equal to 94.
\n
Constraints:
\n\n1 <= n == grid.length <= 100n == grid[i].length0 <= grid[i][j] <= 109给你一个大小为 n x n 的二维矩阵 grid ，一开始所有格子都是白色的。一次操作中，你可以选择任意下标为 (i, j) 的格子，并将第 j 列中从最上面到第 i 行所有格子改成黑色。
如果格子 (i, j) 为白色，且左边或者右边的格子至少一个格子为黑色，那么我们将 grid[i][j] 加到最后网格图的总分中去。
请你返回执行任意次操作以后，最终网格图的 最大 总分数。
\n\n\n\n
示例 1：
\n\n输入：grid = [[0,0,0,0,0],[0,0,3,0,0],[0,1,0,0,0],[5,0,0,3,0],[0,0,0,0,2]]
\n\n输出：11
\n\n解释：
\n\n第一次操作中，我们将第 1 列中，最上面的格子到第 3 行的格子染成黑色。第二次操作中，我们将第 4 列中，最上面的格子到最后一行的格子染成黑色。最后网格图总分为 grid[3][0] + grid[1][2] + grid[3][3] 等于 11 。
示例 2：
\n\n输入：grid = [[10,9,0,0,15],[7,1,0,8,0],[5,20,0,11,0],[0,0,0,1,2],[8,12,1,10,3]]
\n\n输出：94
\n\n解释：
\n\n我们对第 1 ，2 ，3 列分别从上往下染黑色到第 1 ，4， 0 行。最后网格图总分为 grid[0][0] + grid[1][0] + grid[2][1] + grid[4][1] + grid[1][3] + grid[2][3] + grid[3][3] + grid[4][3] + grid[0][4] 等于 94 。
\n\n
提示：
\n\n1 <= n == grid.length <= 100n == grid[i].length0 <= grid[i][j] <= 109dp[i][lastHeight][beforeLastHeight] denote the maximum score if the grid was limited to column i, and the height of column i - 1 is lastHeight and the height of column i - 2 is beforeLastHeight.",
"The third state, beforeLastHeight, is used to determine which values of column i - 1 will be added to the score. We can replace this state with another state that only takes two values 0 or 1.",
"Let dp[i][lastHeight][isBigger] denote the maximum score if the grid was limited to column i, and where the height of column i - 1 is lastHeight. Additionally, if isBigger == 1, the number of black cells in column i is assumed to be larger than the number of black cells in column i - 2, and vice versa. Note that if our assumption is wrong, it would lead to a suboptimal score and, therefore, it would not be considered as the final answer."
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