Count White Balls After K Steps

You are given a ring of `N` positions indexed from `0` to `N-1`. - At each position `i`, there is initially one ball whose color is either white or black. - Each position may also contain a flip gate, represented by `gate[i]` where `1` means a gate is present and `0` means no gate. - In one synchronous step, every ball moves one position clockwise. - Whenever a ball crosses a position containing a flip gate, its color toggles. For each query, you are given an integer `K` and must determine how many balls are white after exactly `K` steps. Additional details: - A ball starting at index `i` will be located at `(i + K) mod N` after `K` steps. - Its final color depends only on whether it crosses an even or odd number of gates during those `K` moves. - `K` may be much larger than `N`, so simulating step by step is too slow. Design an algorithm that preprocesses the ring and answers each query efficiently, ideally in `O(1)` time per query after preprocessing.