Solve powers, phases, grid pops, and swaps

Solve the following four problems: 1) Count powers of k in an array: Given an array of positive integers nums and an integer k >= 1, return how many elements x in nums are exact powers of k (i.e., there exists t >= 0 with x = k^t). Handle the special case k = 1 (only x = 1 qualifies). 2) Compute a moon phase by date: On a planet where moon phases repeat every 8 days (phase indices 0.. 7), you are given the phase index on day 1 of the year and the calendar for that year as a list of month lengths. Given a target date (month, day), compute the phase index on that date by converting the date to a day offset from day 1 and applying modulo 8. Be careful with month-length arithmetic and off-by-one errors. 3) Simulate bubble explosions with gravity: Given an R x C grid of integers representing colors and a threshold T (e.g., T = 3), repeatedly remove any 4-directionally connected component of size >= T consisting of the same color by setting its cells to 0; after each removal, apply gravity within each column so cells fall to the lowest available positions. Continue until no more removals are possible and return the final grid. 4) Count near-equal pairs via at most two swaps: You are given an array of N numeric strings of equal length L (leading zeros allowed). Count unordered pairs (i, j) such that s[i] can be transformed into s[j] using at most two swaps of characters within a single string. Clearly define the transformation rule, state necessary conditions (e.g., matching digit multisets), and design an efficient algorithm with time and space complexity analysis.