Museum Zone Boundary Atlas

A museum has a straight corridor of rooms. Room k has a nonnegative conservation exposure score exposure[k].

The curator wants to divide all rooms into three nonempty, contiguous zones, in their original order. A division is described by two room counts i and j:

• The first zone contains rooms with indices 0 through i - 1.
• The middle zone contains rooms with indices i through j - 1.
• The last zone contains rooms with indices j through n - 1.

Let their total exposure scores be A, B, and C. A division is acceptable only when:

• The three zone lengths are at least min_rooms[0], min_rooms[1], and min_rooms[2], respectively.
• B >= first_factor * A.
• C >= last_factor * B.

Build a boundary atlas for every possible first boundary i from 1 through n - 2. Return a list of n - 2 pairs, where the pair at index i - 1 is:

• [smallest_j, largest_j] if at least one acceptable division uses that first boundary;
• [-1, -1] otherwise.

Because exposure scores are nonnegative, every integer between smallest_j and largest_j is also an acceptable second boundary for that i. Boundaries are room counts, not zero-based room indices.

Input: exposure = [1,1,1,1,1,1], min_rooms = [1,1,1], first_factor = 1, last_factor = 1
Output: [[2,3],[4,4],[-1,-1],[-1,-1]]

Input: exposure = [0,0,0,0,0,0,0], min_rooms = [2,2,1], first_factor = 4, last_factor = 3
Output: [[-1,-1],[4,6],[5,6],[6,6],[-1,-1]]

Input: exposure = [8,1,1,1,1], min_rooms = [1,1,1], first_factor = 2, last_factor = 1
Output: [[-1,-1],[-1,-1],[-1,-1]]

An O(n) time solution with O(n) space, including prefix sums and the returned atlas, is expected. Use sufficiently wide arithmetic for multiplied exposure totals.