<p>You are given a <strong>1-indexed</strong> array of integers <code>nums</code> of length <code>n</code>.</p>

<p>We define a function <code>greaterCount</code> such that <code>greaterCount(arr, val)</code> returns the number of elements in <code>arr</code> that are <strong>strictly greater</strong> than <code>val</code>.</p>

<p>You need to distribute all the elements of <code>nums</code> between two arrays <code>arr1</code> and <code>arr2</code> using <code>n</code> operations. In the first operation, append <code>nums[1]</code> to <code>arr1</code>. In the second operation, append <code>nums[2]</code> to <code>arr2</code>. Afterwards, in the <code>i<sup>th</sup></code> operation:</p>

<ul>
	<li>If <code>greaterCount(arr1, nums[i]) &gt; greaterCount(arr2, nums[i])</code>, append <code>nums[i]</code> to <code>arr1</code>.</li>
	<li>If <code>greaterCount(arr1, nums[i]) &lt; greaterCount(arr2, nums[i])</code>, append <code>nums[i]</code> to <code>arr2</code>.</li>
	<li>If <code>greaterCount(arr1, nums[i]) == greaterCount(arr2, nums[i])</code>, append <code>nums[i]</code> to the array with a <strong>lesser</strong> number of elements.</li>
	<li>If there is still a tie, append <code>nums[i]</code> to <code>arr1</code>.</li>
</ul>

<p>The array <code>result</code> is formed by concatenating the arrays <code>arr1</code> and <code>arr2</code>. For example, if <code>arr1 == [1,2,3]</code> and <code>arr2 == [4,5,6]</code>, then <code>result = [1,2,3,4,5,6]</code>.</p>

<p>Return <em>the integer array</em> <code>result</code>.</p>

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

<pre>
<strong>Input:</strong> nums = [2,1,3,3]
<strong>Output:</strong> [2,3,1,3]
<strong>Explanation:</strong> After the first 2 operations, arr1 = [2] and arr2 = [1].
In the 3<sup>rd</sup> operation, the number of elements greater than 3 is zero in both arrays. Also, the lengths are equal, hence, append nums[3] to arr1.
In the 4<sup>th</sup> operation, the number of elements greater than 3 is zero in both arrays. As the length of arr2 is lesser, hence, append nums[4] to arr2.
After 4 operations, arr1 = [2,3] and arr2 = [1,3].
Hence, the array result formed by concatenation is [2,3,1,3].
</pre>

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

<pre>
<strong>Input:</strong> nums = [5,14,3,1,2]
<strong>Output:</strong> [5,3,1,2,14]
<strong>Explanation:</strong> After the first 2 operations, arr1 = [5] and arr2 = [14].
In the 3<sup>rd</sup> operation, the number of elements greater than 3 is one in both arrays. Also, the lengths are equal, hence, append nums[3] to arr1.
In the 4<sup>th</sup> operation, the number of elements greater than 1 is greater in arr1 than arr2 (2 &gt; 1). Hence, append nums[4] to arr1.
In the 5<sup>th</sup> operation, the number of elements greater than 2 is greater in arr1 than arr2 (2 &gt; 1). Hence, append nums[5] to arr1.
After 5 operations, arr1 = [5,3,1,2] and arr2 = [14].
Hence, the array result formed by concatenation is [5,3,1,2,14].
</pre>

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

<pre>
<strong>Input:</strong> nums = [3,3,3,3]
<strong>Output:</strong> [3,3,3,3]
<strong>Explanation:</strong> At the end of 4 operations, arr1 = [3,3] and arr2 = [3,3].
Hence, the array result formed by concatenation is [3,3,3,3].
</pre>

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

<ul>
	<li><code>3 &lt;= n &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= nums[i] &lt;= 10<sup>9</sup></code></li>
</ul>

class FenwickTree:
  def __init__(self, n: int):
    self.sums = [0] * (n + 1)

  def add(self, i: int, delta: int) -> None:
    while i < len(self.sums):
      self.sums[i] += delta
      i += FenwickTree.lowbit(i)

  def get(self, i: int) -> int:
    summ = 0
    while i > 0:
      summ += self.sums[i]
      i -= FenwickTree.lowbit(i)
    return summ

  @staticmethod
  def lowbit(i: int) -> int:
    return i & -i


class Solution:
  def resultArray(self, nums: list[int]) -> list[int]:
    arr1 = []
    arr2 = []
    ranks = self._getRanks(nums)
    tree1 = FenwickTree(len(ranks))
    tree2 = FenwickTree(len(ranks))

    def add(num: int, arr: list[int], tree: FenwickTree) -> None:
      arr.append(num)
      tree.add(ranks[num], 1)

    add(nums[0], arr1, tree1)
    add(nums[1], arr2, tree2)

    for i in range(2, len(nums)):
      greaterCount1 = len(arr1) - tree1.get(ranks[nums[i]])
      greaterCount2 = len(arr2) - tree2.get(ranks[nums[i]])
      if greaterCount1 > greaterCount2:
        add(nums[i], arr1, tree1)
      elif greaterCount1 < greaterCount2:
        add(nums[i], arr2, tree2)
      elif len(arr1) > len(arr2):
        add(nums[i], arr2, tree2)
      else:
        add(nums[i], arr1, tree1)

    return arr1 + arr2

  def _getRanks(self, nums: list[int]) -> dict[int, int]:
    ranks = collections.Counter()
    rank = 0
    for num in sorted(set(nums)):
      rank += 1
      ranks[num] = rank
    return ranks
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