Painter Partition Problem

You have a list of board lengths `boards` (positive integers) and exactly `k` painters. Each painter must paint one contiguous sequence of boards, and every painter paints at least one board. The time a painter takes equals the sum of the lengths of the boards assigned to that painter. The overall job is finished when all painters finish, so the total time is the maximum sum among the painters. Write a function `min_largest_sum(boards: list[int], k: int) -> int` that returns the minimum possible maximum sum when you partition the list into `k` contiguous non-empty segments. **Examples** - `min_largest_sum([3, 2, 4], 2)` returns `5` (partition as [3,2] and [4]). - `min_largest_sum([10, 20, 30, 40], 2)` returns `60` (partition as [10,20,30] and [40] or [10] and [20,30,40] gives max 60; other splits give 70 or 80). **Assumptions** - `boards` has at least 1 element. - `1 <= k <= len(boards)`. - Each board length is a positive integer. - You may assume that all inputs satisfy these bounds; no need to validate.

>>> min_largest_sum([3, 2, 4], 2)
5
>>> min_largest_sum([10, 20, 30, 40], 2)
60
>>> min_largest_sum([1, 2, 3, 4, 5], 3)
6
>>> min_largest_sum([5, 5, 5, 5], 4)
5