Delete a Position from Interval Coverage

Question

A compact coverage set is represented by sorted, non-overlapping, non-touching inclusive intervals. If the covered values were expanded into one ordered sequence, delete the value at a given zero-based position and return the updated compact intervals.

For example, `[(4, 7), (10, 11), (13, 15)]` represents `4,5,6,7,10,11,13,14,15`. Deleting position `2` removes value `6`, producing `[[4, 5], [7, 7], [10, 11], [13, 15]]`.

```python
def remove_cover(intervals: list, index: int) -> list: ...
```

## Requirements

- Return intervals as `[start, end]` pairs.
- Preserve sorted, non-overlapping, non-touching output.
- A deletion may shrink an endpoint, remove a singleton interval, or split an interval in two.
- Assume `index` refers to an existing covered value; state how you would report an invalid index.
- Do not expand every covered integer when ranges may be large.
- Return newly allocated pairs without mutating the input. Bounds and index use integer arithmetic; Python arbitrary-precision integers are permitted.

## Design discussion (not graded)

1. Support a sequence of deletions efficiently.
2. Delete a range of positions rather than one position.
3. Design a structure supporting insertion, deletion, and selection by covered-position rank.
4. Analyze integer-overflow risks when interval lengths are very large.