Minimum Time to Transport All Individuals

Description

You are given n individuals at a base camp who need to cross a river to reach a destination using a single boat. The boat can carry at most k people at a time. The trip is affected by environmental conditions that vary cyclically over m stages.

Each stage j has a speed multiplier mul[j]:

- If mul[j] > 1, the trip slows down.

- If mul[j] < 1, the trip speeds up.

Each individual i has a rowing strength represented by time[i], the time (in minutes) it takes them to cross alone in neutral conditions.

Rules:

- A group g departing at stage j takes time equal to the maximum time[i] among its members, multiplied by mul[j] minutes to reach the destination.

- After the group crosses the river in time d, the stage advances by floor(d) % m steps.

- If individuals are left behind, one person must return with the boat. Let r be the index of the returning person, the return takes time[r] × mul[current_stage], defined as return_time, and the stage advances by floor(return_time) % m.

Return the minimum total time required to transport all individuals. If it is not possible to transport all individuals to the destination, return -1.