Alice had a 0-indexed array `arr` consisting of `n` positive integers. She chose an arbitrary positive integer `k` and created two new 0-indexed integer arrays `lower` and `higher` in the following manner:

1. `lower[i] = arr[i] - k`, for every index `i` where `0 <= i < n`
2. `higher[i] = arr[i] + k`, for every index `i` where `0 <= i < n`

Unfortunately, Alice lost all three arrays. However, she remembers the integers that were present in the arrays `lower` and `higher`, but not the array each integer belonged to. Help Alice and recover the original array.

Given an array `nums` consisting of `2n` integers, where exactly `n` of the integers were present in `lower` and the remaining in `higher`, return the original array `arr`. In case the answer is not unique, return any valid array.

Note: The test cases are generated such that there exists at least one valid array `arr`.

Example 1:

    Input: nums = [2,10,6,4,8,12]
    Output: [3,7,11]
    Explanation:
    If arr = [3,7,11] and k = 1, we get lower = [2,6,10] and higher = [4,8,12].
    Combining lower and higher gives us [2,6,10,4,8,12], which is a permutation of nums.
    Another valid possibility is that arr = [5,7,9] and k = 3. In that case, lower = [2,4,6] and higher = [8,10,12].

Example 2:

    Input: nums = [1,1,3,3]
    Output: [2,2]
    Explanation:
    If arr = [2,2] and k = 1, we get lower = [1,1] and higher = [3,3].
    Combining lower and higher gives us [1,1,3,3], which is equal to nums.
    Note that arr cannot be [1,3] because in that case, the only possible way to obtain [1,1,3,3] is with k = 0.
    This is invalid since k must be positive.

Example 3:

    Input: nums = [5,435]
    Output: [220]
    Explanation:
    The only possible combination is arr = [220] and k = 215. Using them, we get lower = [5] and higher = [435].

Constraints:

- `2 * n == nums.length`
- `1 <= n <= 1000`
- `1 <= nums[i] <= 109`
- The test cases are generated such that there exists at least one valid array `arr`.