Find the Period of a Perfect Interleaving

### Problem Begin with the identity permutation of even length `n`. In each operation create a new array: at even index i take the previous entry at i/2; at odd index i take the previous entry at n/2+(i-1)/2. Replace the old permutation simultaneously. Return the least positive operation count that restores the identity. ```cpp class Solution { public: int reinitializePermutation(int n); }; ``` ### Examples **Example 1:** ```text n = 8 Output: 3 Explanation: Interior position 1 moves through 2, 4, and back to 1 modulo 7. ``` **Example 2:** ```text n = 10 Output: 6 Explanation: Position 1 follows 1 -> 2 -> 4 -> 8 -> 7 -> 5 -> 1 modulo 9. ``` ### Constraints - `2 <= n <= 1000`. - n is even. ### Clarifying Questions The update reads the old permutation for every new entry. The answer must be positive even if the first operation leaves the permutation unchanged. ```hint Inspect the structure Does each new entry read the old permutation, and must the reported restoration count be positive? ``` ### Reasoning checkpoints - Interpret the simultaneous old-to-new permutation rule without mixing movement and lookup directions. - Prove the least positive return time for the entire permutation, including `n = 2`. ### Optional extensions How would direct array simulation compare with tracking a single interior position?