Minimum Time to Transport All Individuals

Schedule at-most-k-person crossings and single-person returns to get all n people across while minimizing total time, where each trip takes max(time of group) × mul[current_stage] and the stage advances by floor(duration) % m, producing non-linear, discrete state transitions. The core challenge is optimizing the sequence of group choices and returners under capacity and cyclic-stage effects (typically solved with DP/shortest-path over subsets and current stage).