Compose a Zero-Based Permutation with Itself

### Problem Given a zero-based permutation `nums` of length `n`, return an array `ans` of the same length satisfying `ans[i] = nums[nums[i]]` for every index. Both lookups refer to the original permutation. The source follow-up asks for an approach using `O(1)` auxiliary memory. ```cpp class Solution { public: vector<int> buildArray(vector<int>& nums); }; ``` ### Examples **Example 1:** ```text nums = [1,2,0] Output: [2,0,1] Explanation: Looking up indices 1, 2, 0 yields 2, 0, 1. ``` **Example 2:** ```text nums = [2,1,0,3] Output: [0,1,2,3] Explanation: Composing these two swaps and fixed points gives the identity. ``` ### Constraints - `1 <= nums.length <= 1000`. - Entries are distinct and lie in `[0,nums.length-1]`. ### Clarifying Questions Both lookups use original values. The source does not require preserving the input array. ```hint Inspect the structure Can one integer store both an old value and a new value when both are smaller than n? ``` ### Reasoning checkpoints - Produce every requested double lookup using original values, even when storage is reused. - For the source space follow-up, justify the memory accounting and the intermediate integer range. ### Optional extensions The source follow-up asks whether the result can be produced with `O(1)` auxiliary memory; explain which storage is excluded as output.