<p>Given an integer array <code>nums</code> and two integers <code>lower</code> and <code>upper</code>, return <em>the number of range sums that lie in</em> <code>[lower, upper]</code> <em>inclusive</em>.</p>

<p>Range sum <code>S(i, j)</code> is defined as the sum of the elements in <code>nums</code> between indices <code>i</code> and <code>j</code> inclusive, where <code>i &lt;= j</code>.</p>

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

<pre>
<strong>Input:</strong> nums = [-2,5,-1], lower = -2, upper = 2
<strong>Output:</strong> 3
<strong>Explanation:</strong> The three ranges are: [0,0], [2,2], and [0,2] and their respective sums are: -2, -1, 2.
</pre>

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

<pre>
<strong>Input:</strong> nums = [0], lower = 0, upper = 0
<strong>Output:</strong> 1
</pre>

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

<ul>
	<li><code>1 &lt;= nums.length &lt;= 10<sup>5</sup></code></li>
	<li><code>-2<sup>31</sup> &lt;= nums[i] &lt;= 2<sup>31</sup> - 1</code></li>
	<li><code>-10<sup>5</sup> &lt;= lower &lt;= upper &lt;= 10<sup>5</sup></code></li>
	<li>The answer is <strong>guaranteed</strong> to fit in a <strong>32-bit</strong> integer.</li>
</ul>

from typing import List


class Solution:
    def countRangeSum(self, nums: List[int], lower: int, upper: int) -> int:
        prefix = [0]
        sum = 0
        for num in nums:
            sum += num
            prefix.append(sum)
        return self.merge(prefix, 0, len(prefix), lower, upper)
        
    def merge(self, prefix, lo, hi, lower, upper):
        mid = (lo+hi) // 2
        if mid == lo:
            return 0
        count = self.merge(prefix, lo, mid, lower, upper) + self.merge(prefix, mid, hi, lower, upper) 
        i = j = mid
        for left in prefix[lo:mid]:
            while i < hi and prefix[i] - left <  lower: i += 1
            while j < hi and prefix[j] - left <= upper: j += 1
            count += j - i
        prefix[lo:hi] = sorted(prefix[lo:hi])
        return count
    
31701