<p>Given an integer array <code>nums</code>, return <em>the number of <strong>reverse pairs</strong> in the array</em>.</p>

<p>A <strong>reverse pair</strong> is a pair <code>(i, j)</code> where:</p>

<ul>
	<li><code>0 &lt;= i &lt; j &lt; nums.length</code> and</li>
	<li><code>nums[i] &gt; 2 * nums[j]</code>.</li>
</ul>

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

<pre>
<strong>Input:</strong> nums = [1,3,2,3,1]
<strong>Output:</strong> 2
<strong>Explanation:</strong> The reverse pairs are:
(1, 4) --&gt; nums[1] = 3, nums[4] = 1, 3 &gt; 2 * 1
(3, 4) --&gt; nums[3] = 3, nums[4] = 1, 3 &gt; 2 * 1
</pre>

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

<pre>
<strong>Input:</strong> nums = [2,4,3,5,1]
<strong>Output:</strong> 3
<strong>Explanation:</strong> The reverse pairs are:
(1, 4) --&gt; nums[1] = 4, nums[4] = 1, 4 &gt; 2 * 1
(2, 4) --&gt; nums[2] = 3, nums[4] = 1, 3 &gt; 2 * 1
(3, 4) --&gt; nums[3] = 5, nums[4] = 1, 5 &gt; 2 * 1
</pre>

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

<ul>
	<li><code>1 &lt;= nums.length &lt;= 5 * 10<sup>4</sup></code></li>
	<li><code>-2<sup>31</sup> &lt;= nums[i] &lt;= 2<sup>31</sup> - 1</code></li>
</ul>

class Solution:
    @staticmethod
    def count_reverse(target: int, sorted_integers: list[int] | tuple[int]):
        if not sorted_integers:
            return 0, 0

        insertion_idx = 0  # Target's idx into sorted integers to maintain order.
        back_idx, front_idx = 0, len(sorted_integers) - 1

        while back_idx <= front_idx:  # First while: search for insertion idx.
            mid_idx = (back_idx + front_idx) // 2
            if sorted_integers[mid_idx] < target:
                back_idx = mid_idx + 1
                continue
            front_idx = mid_idx - 1

        insertion_idx = back_idx

        back_idx, front_idx = 0, len(sorted_integers) - 1
        while back_idx <= front_idx:  # Second while: count number of ints < 0.5 * target.
            mid_idx = (back_idx + front_idx) // 2
            if 2 * sorted_integers[mid_idx] < target:
                back_idx = mid_idx + 1
                continue
            front_idx = mid_idx - 1

        return back_idx, insertion_idx  # Back idx implies number of ints < 0.5 * target.

    def reversePairs(self, integers: List[int]) -> int:
        if len(integers) <= 1:
            return 0

        reverse_pairs = 0
        sorted_integers = [integers.pop(-1)]  # Start from rightmost int.
        while integers:
            popped_integer = integers.pop(-1)
            reverse_count, insertion_idx = self.count_reverse(popped_integer, sorted_integers)
            reverse_pairs += reverse_count
            sorted_integers.insert(insertion_idx, popped_integer)

        return reverse_pairs

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