You are given a 2D integer matrix grid of size n x m, an integer array limits of length n, and an integer k. The task is to find the maximum sum of at most k elements from the matrix grid such that:

The number of elements taken from the i^th row of grid does not exceed limits[i].

Return the maximum sum.

Example 1:

Input: grid = [[1,2],[3,4]], limits = [1,2], k = 2

Output: 7

Explanation:

From the second row, we can take at most 2 elements. The elements taken are 4 and 3.
The maximum possible sum of at most 2 selected elements is 4 + 3 = 7.

Example 2:

Input: grid = [[5,3,7],[8,2,6]], limits = [2,2], k = 3

Output: 21

Explanation:

From the first row, we can take at most 2 elements. The element taken is 7.
From the second row, we can take at most 2 elements. The elements taken are 8 and 6.
The maximum possible sum of at most 3 selected elements is 7 + 8 + 6 = 21.

Constraints:

n == grid.length == limits.length
m == grid[i].length
1 <= n, m <= 500
0 <= grid[i][j] <= 10⁵
0 <= limits[i] <= m
0 <= k <= min(n * m, sum(limits))