Egg Drop Puzzle

You are given `k` eggs and a building with `n` floors (numbered 1 to n). There exists a critical floor `f` (0 <= f <= n) such that eggs dropped from floors <= f do not break, and eggs dropped from floors > f break. You need to determine the exact value of `f` in the worst case with the fewest number of egg drops. Implement the function `min_drops(k: int, n: int) -> int` that returns the minimum number of drops required to guarantee finding the critical floor. **Rules:** - An egg that survives a drop can be reused. - An egg that breaks is discarded and cannot be reused. - You have at most `k` eggs initially. - If you have only one egg, you must test floors sequentially from lowest to highest (worst-case drops = n). - If you have unlimited eggs, the problem reduces to binary search, but with limited eggs you must balance. **Input:** Two integers `k` and `n`, where `1 <= k <= 100` and `1 <= n <= 10000`. Both are within these bounds. The answer fits in a 32-bit integer. **Output:** An integer representing the minimum number of drops needed in the worst case. **Complexity:** Your solution should be efficient for the given bounds. An O(k n) dynamic programming approach is acceptable.

>>> min_drops(1, 2)
2
>>> min_drops(2, 6)
3
>>> min_drops(3, 14)
4
>>> min_drops(2, 100)
14