List Point Pairs Spanning Non-Degenerate Boxes in N Dimensions, Sorted by Volume

You are given `n` points in `d`-dimensional space; point `i` (0-based) has integer coordinates `points[i]`. Any two points `i < j` define an axis-aligned box that has the two points as opposite corners. The box's volume is the product, over all `d` axes, of the absolute difference between the two points' coordinates on that axis. In two dimensions the box is a rectangle and its volume is the rectangle's area; the same function must work for every dimension `d` in the stated range. Return every pair of points whose box has a non-zero volume, together with that volume. ### Function Signature ```python def box_volumes(points: List[List[int]]) -> List[List[int]]: ``` ### Rules - Each reported pair is a list `[i, j, volume]` with `i < j`, where `i` and `j` are indices into `points`. - A pair whose volume is 0 (the two points share a coordinate on at least one axis, including identical points) is not reported. - Order the result by volume descending; break ties by `i` ascending, then by `j` ascending. - Return an empty list when no pair qualifies. ### Constraints - `1 <= n <= 300` - `2 <= d <= 6`, and every point has exactly `d` coordinates - `-100 <= points[i][k] <= 100` - Every volume is at most `200^6 = 6.4 * 10^13`, which fits in a 64-bit integer and stays below `2^53` ### Examples **Example 1** ``` points = [[1, 1], [4, 3], [1, 5], [3, 0]] output = [[2, 3, 10], [0, 1, 6], [1, 2, 6], [1, 3, 3], [0, 3, 2]] ``` Points 0 and 2 share the x-coordinate 1, so their rectangle has zero area and is left out. Pairs (0, 1) and (1, 2) both have area 6 and are ordered by their first index. **Example 2** ``` points = [[0, 0, 0], [2, 3, 1], [2, 1, 4]] output = [[0, 2, 8], [0, 1, 6]] ``` Points 1 and 2 share the first coordinate, so their box is flat and is left out. **Example 3** ``` points = [[5, 5]] output = [] ```