# Validate a Hierarchy and Compute Propagation Time Implement `hierarchy_propagation_time(manager: list[int], inform_time: list[int]) -> int`. Employee `i` reports to `manager[i]`. Exactly one root should have manager `-1`. When an employee receives a message, that employee takes `inform_time[i]` time units before all direct reports receive it. The root has the message at time zero. If the manager relation is not one rooted hierarchy because it has no root, multiple roots, an out-of-range parent, a self-parent, a cycle, or a node unreachable from the root, return `-1`. Otherwise, return the earliest time by which every employee has received the message. A leaf's own `inform_time` does not add time after that leaf receives the message. ## Valid Input Domain - The two arrays have equal length. - Every `inform_time[i]` is a nonnegative integer. ## Constraints - `1 <= manager.length <= 200,000` - `0 <= inform_time[i] <= 1,000,000` - The answer fits in a signed 64-bit integer. ## Public Examples ### Example 1 Input: `manager = [-1, 0, 0, 1], inform_time = [2, 3, 0, 0]` Output: `5` Employee `3` receives the message after the root's two units and employee `1`'s three units. ### Example 2 Input: `manager = [1, 0], inform_time = [1, 1]` Output: `-1` There is no root and the two employees form a cycle. ```hint Separate validity from timing Ensure that every node belongs to one acyclic rooted structure while accumulating the time along root-to-node paths. ```