Holiday Contest 2025 - Guts Round - Set 8 Problem 1

All 705 members in the TOPSOJ discord server decided to go to a party! They went to a mansion with 100 rooms, and each person went to a random room. We label the rooms 1,2,\dots, 100 (where room i has x_i people) such that if i < j, then x_i \le x_j. Every minute, some member in room i moves to a room with at least x_i people (not including the member who just moved). Then, the rooms are relabeled as described above. Now, at first, room i has a_i people in it. After K minutes, room i has b_i people in it. If K is maximal, find \sum_{i=1}^{100}|a_i - b_i|.