Partition to k subsets

Write a function `can_partition_k_subsets(nums: list[int], k: int) -> bool` that returns `True` if the list of positive integers `nums` can be partitioned into `k` subsets, each with the same sum. If not possible, return `False`. Each element of `nums` must be used in exactly one subset. The order of elements does not matter. You may assume `k` is a positive integer and `nums` may be empty. If `nums` is empty, the only way to partition it into `k` subsets is if `k == 0`. Since `k` is positive, return `False` for an empty list. Examples: - `nums = [4,3,2,3,5,2,1]`, `k = 4` → returns `True` because the subsets are `[5], [1,4], [2,3], [2,3]`. - `nums = [1,2,3,4]`, `k = 3` → returns `False` because the total sum 10 is not divisible by 3. - `nums = [1,2,3,4]`, `k = 2` → returns `True` because `[1,4]` and `[2,3]` each sum to 5.

>>> can_partition_k_subsets([4,3,2,3,5,2,1], 4)
True
>>> can_partition_k_subsets([1,2,3,4], 3)
False
>>> can_partition_k_subsets([1,2,3,4], 2)
True
>>> can_partition_k_subsets([], 1)
False