Solve six algorithmic problems

Answer the following independent algorithmic prompts. For each, explain your approach, justify data structures, analyze time/space complexity, and provide correct code:

1. Bulb toggling passes: You have n bulbs initially off. For pass i (1 ≤ i ≤ n), toggle every bulb whose index is a multiple of i. After n passes, return the count of bulbs that are on. Follow-up: given m ≤ n passes or a custom toggling set S of divisors, compute the result efficiently without simulating all passes.
2. Repeated word-distance queries: Preprocess an array of words to support many queries of the form shortestDistance(word1, word
3. that returns the minimum index gap between any occurrence of the two words. Design the initialization and query APIs, then optimize for both time and memory. Follow-ups: support dynamic inserts/appends; support queries over a sliding window.
4. Invert a binary tree: Given the root of a binary tree, return its mirror image. Discuss iterative vs. recursive solutions and how to handle nodes missing exactly one child. Follow-ups: verify structural equality between the original and double-inverted tree; invert only nodes at odd depths.
5. Count land clusters in a grid: Given an m×n grid of '1' (land) and '0' (water), return the number of connected land clusters using 4-directional adjacency. Follow-ups: handle dynamic updates turning cells on/off; extend to 8-directional adjacency; optimize with union–find.
6. Longest run in a circular binary array: Given a binary array treated as circular (index wraps around), compute the maximum number of consecutive 1s. Follow-ups: if you may flip up to k zeros to ones, find the new maximum and return one optimal window.
7. Peel leaves by rounds: Given a binary tree, iteratively collect and remove its leaves; output a list of lists where the i-th list contains the values removed at round i. Provide an O(n) solution and discuss how computing node heights leads to the result.