2D Max Pooling That Also Returns the Coordinates of Each Window Maximum

Implement 2D max pooling over a single-channel grid of integers. A `k`-by-`k` window slides over the grid with step `stride` in both directions, and each window produces the maximum value inside it. As the follow-up, also report where each maximum came from: for every output cell, return the maximum value together with its row and column in the input grid. ### Function Signature ```python def max_pool_with_argmax(grid: list[list[int]], k: int, stride: int) -> list[list[list[int]]]: ``` ### Rules - No padding is used. A window is placed with its top-left corner at `(i * stride, j * stride)` for every `i, j >= 0` such that the whole window fits inside the grid. With `R` rows and `C` columns, the output has `(R - k) // stride + 1` rows and `(C - k) // stride + 1` columns. Windows that would extend past the grid edge are not produced. - Output cell `[i][j]` is the list `[max_value, row, col]`, where `max_value` is the largest value in window `(i, j)` and `(row, col)` is its position in `grid`. - If the maximum occurs more than once in a window, report the occurrence with the smallest row, and among those the smallest column. - If `k` is larger than the number of rows or columns, no window fits: return an empty list `[]`. ### Constraints - `1 <= len(grid) <= 100` and `1 <= len(grid[r]) <= 100`; all rows have the same length. - `-10^9 <= grid[r][c] <= 10^9` - `1 <= k <= 100` - `1 <= stride <= 100` ### Examples **Example 1** - Input: `grid = [[1, 3, 2, 0], [4, 6, 5, 1], [7, 2, 9, 8], [3, 4, 1, 6]]`, `k = 2`, `stride = 2` - Output: `[[[6, 1, 1], [5, 1, 2]], [[7, 2, 0], [9, 2, 2]]]` - Explanation: The four non-overlapping 2-by-2 windows have maxima 6 at `(1, 1)`, 5 at `(1, 2)`, 7 at `(2, 0)` and 9 at `(2, 2)`. **Example 2** - Input: `grid = [[5, 1, 5], [2, 5, 0], [1, 1, 3]]`, `k = 2`, `stride = 1` - Output: `[[[5, 0, 0], [5, 0, 2]], [[5, 1, 1], [5, 1, 1]]]` - Explanation: The top-left window contains 5 at `(0, 0)` and at `(1, 1)`; the smaller row wins. The top-right window contains 5 at `(0, 2)` and `(1, 1)`; row 0 wins. **Example 3** - Input: `grid = [[1, 2], [3, 4]]`, `k = 3`, `stride = 1` - Output: `[]`