Minimize total factory cost with distance penalties

You must choose construction options for factories to minimize a total cost. Input format: - `options[i]` is a list of options for factory `i`. - Each option is a pair `[cost, distance]`. For 3 factories, an example input is: ```python options = [ [[10, 0], [20, 0], [35, 0]], [[35, 0], [50, 0], [25, 0]], [[30, 0], [5, 0], [40, 0]] ] ``` ## Total cost (for N factories) If you pick exactly one option per built factory, define: - Build cost = sum of chosen `cost`. - Distance penalty = sum over adjacent built factories of `abs(distance[i] - distance[i+1])`. Total = build cost + distance penalty. ## Questions 1. **3 factories, ignore distance**: All distances are 0 (or distance penalties are ignored). Compute the minimum possible total cost. 2. **3 factories, include distance**: Use the full total cost formula. Compute the minimum. 3. **N factories, include distance**: Generalize to any `N` factories. Compute the minimum efficiently. 4. **N factories, must skip exactly one factory**: You must choose options for exactly `N-1` factories, skipping (not building) exactly one factory. Distance penalty applies between adjacent built factories in their original order (i.e., skipping connects its neighbors). Compute the minimum. ## Output Return the minimum total cost (and optionally the chosen options if requested). ## Constraints / Notes - Assume `N` and number of options per factory can be large enough that brute force over all combinations may be too slow. - Discuss time/space complexity tradeoffs.