## Count Rectangle Coverage on a Grid You are given an `n x n` grid initially filled with zeroes and a list of inclusive rectangular updates. Each rectangle adds one to every covered cell. Implement: ```text range_add_counts(n, rectangles) -> list[list[int]] ``` Each rectangle is `[row1, col1, row2, col2]`, using zero-based inclusive coordinates. Return the final grid, where each cell contains the number of rectangles covering it. ### Example ```text n = 3 rectangles = [ [0, 0, 1, 1], [1, 1, 2, 2] ] result = [ [1, 1, 0], [1, 2, 1], [0, 1, 1] ] ``` Cell `(1, 1)` is covered by both rectangles. ### Constraints - `1 <= n <= 500` - `0 <= len(rectangles) <= 100000` - `0 <= row1 <= row2 < n` - `0 <= col1 <= col2 < n` - Duplicate rectangles count as separate updates. - If `rectangles` is empty, return an all-zero grid. ### Clarifications - Both lower and upper rectangle coordinates are included. - The input list must not be mutated. - Returning a newly allocated matrix is expected. ### Hints - Updating every cell of every rectangle is too slow at the upper bound. - A two-dimensional difference array can encode a rectangle with four boundary changes. - Recover cell values by taking prefix sums across both dimensions, taking care at the grid edges. ### Discussion Extensions - Derive the time and space complexity. - How would you adapt the method if coordinates were very large but only a small number of points were queried? - How would the boundary updates change for half-open rectangles?