<p>You are given an integer <code>n</code> representing the length of an unknown array that you are trying to recover. You are also given an array <code>sums</code> containing the values of all <code>2<sup>n</sup></code> <strong>subset sums</strong> of the unknown array (in no particular order).</p>

<p>Return <em>the array </em><code>ans</code><em> of length </em><code>n</code><em> representing the unknown array. If <strong>multiple</strong> answers exist, return <strong>any</strong> of them</em>.</p>

<p>An array <code>sub</code> is a <strong>subset</strong> of an array <code>arr</code> if <code>sub</code> can be obtained from <code>arr</code> by deleting some (possibly zero or all) elements of <code>arr</code>. The sum of the elements in <code>sub</code> is one possible <strong>subset sum</strong> of <code>arr</code>. The sum of an empty array is considered to be <code>0</code>.</p>

<p><strong>Note:</strong> Test cases are generated such that there will <strong>always</strong> be at least one correct answer.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> n = 3, sums = [-3,-2,-1,0,0,1,2,3]
<strong>Output:</strong> [1,2,-3]
<strong>Explanation: </strong>[1,2,-3] is able to achieve the given subset sums:
- []: sum is 0
- [1]: sum is 1
- [2]: sum is 2
- [1,2]: sum is 3
- [-3]: sum is -3
- [1,-3]: sum is -2
- [2,-3]: sum is -1
- [1,2,-3]: sum is 0
Note that any permutation of [1,2,-3] and also any permutation of [-1,-2,3] will also be accepted.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> n = 2, sums = [0,0,0,0]
<strong>Output:</strong> [0,0]
<strong>Explanation:</strong> The only correct answer is [0,0].
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> n = 4, sums = [0,0,5,5,4,-1,4,9,9,-1,4,3,4,8,3,8]
<strong>Output:</strong> [0,-1,4,5]
<strong>Explanation:</strong> [0,-1,4,5] is able to achieve the given subset sums.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= n &lt;= 15</code></li>
	<li><code>sums.length == 2<sup>n</sup></code></li>
	<li><code>-10<sup>4</sup> &lt;= sums[i] &lt;= 10<sup>4</sup></code></li>
</ul>

from collections import Counter
from typing import List

class Solution:
    def recoverArray(self, n: int, sums: List[int]) -> List[int]:
        sums.sort()
        
        def recover(n, sums):
            if n == 0:
                return []
            
            # The smallest element `x` should be the difference between the smallest two elements
            diff = sums[1] - sums[0]
            
            # Use a counter to split the sums into two groups
            counter = Counter(sums)
            sums_with_x = []
            sums_without_x = []
            
            for s in sums:
                if counter[s] > 0:
                    sums_without_x.append(s)
                    counter[s] -= 1
                    if counter[s + diff] > 0:
                        sums_with_x.append(s + diff)
                        counter[s + diff] -= 1
            
            # Determine if the diff should be added or subtracted
            # Check if the zero is in the group without `diff`
            if 0 in sums_without_x:
                return [diff] + recover(n - 1, sums_without_x)
            else:
                return [-diff] + recover(n - 1, sums_with_x)
        
        return recover(n, sums)

# Example usage:
# sol = Solution()
# print(sol.recoverArray(3, [-3,-2,-1,0,0,1,2,3]))  # Output: [1, 2, -3]
# print(sol.recoverArray(2, [0,0,0,0]))  # Output: [0, 0]
# print(sol.recoverArray(4, [0,0,5,5,4,-1,4,9,9,-1,4,3,4,8,3,8]))  # Output: [0, -1, 4, 5]
22400