<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>

class Solution:
  def recoverArray(self, n: int, sums: list[int]) -> list[int]:
    def recover(sums: list[int]) -> list[int]:
      if len(sums) == 1:
        return []

      count = collections.Counter(sums)
      # Either num or -num must be in the final array.
      #  num + sumsExcludingNum = sumsIncludingNum
      # -num + sumsIncludingNum = sumsExcludingNum
      num = sums[1] - sums[0]
      sumsExcludingNum = []
      sumsIncludingNum = []
      chooseSumsExcludingNum = True

      for summ in sums:
        if count[summ] == 0:
          continue
        count[summ] -= 1
        count[summ + num] -= 1
        sumsExcludingNum.append(summ)
        sumsIncludingNum.append(summ + num)
        if summ + num == 0:
          chooseSumsExcludingNum = False

      # Choose `sumsExludingNum` by default since we want to gradually strip
      # `num` from each sum in `sums` to have the final array. However, we should
      # always choose the group of sums with 0 since it's a must-have.
      return ([num] + recover(sumsExcludingNum) if chooseSumsExcludingNum else
              [-num] + recover(sumsIncludingNum))

    return recover(sorted(sums))
1168