<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 __init__(self):
        self.ans = 0

    def add(self, a, b):
        j, k = 0, 0
        while j < len(a) and k < len(b):
            if a[j] < b[k]:
                j += 1
            else:
                self.ans += len(a) - j
                k += 1

    def merge(self, a, b):
        c = []
        j, k = 0, 0
        while j < len(a) and k < len(b):
            if a[j] < b[k]:
                c.append(a[j])
                j += 1
            else:
                self.ans += len(a) - j
                c.append(b[k])
                k += 1
        while j < len(a):
            c.append(a[j])
            j += 1
        while k < len(b):
            c.append(b[k])
            k += 1
        return c

    def sorts(self, a, x):
        if len(a) <= 1:
            return a
        n = len(a)
        b = a[:n//2]
        c = a[n//2:]
        
        xx = self.sorts(b, x)
        yy = self.sorts(c, x)
        
        d = [val - x for val in yy]
        self.add(xx, d)
        
        return self.merge(xx, yy)

    def func(self, a, x):
        self.ans = 0
        self.sorts(a, x)
        return self.ans

    def countRangeSum(self, nums: List[int], lower: int, upper: int) -> int:
        num = [0] * len(nums)
        for i in range(len(nums)):
            num[i] = nums[i]
        
        n = len(nums)
        for i in range(1, n):
            num[i] += num[i - 1]

        key1 = self.func(num, lower - 1)
        key2 = self.func(num, upper)
        
        key3 = sum(1 for val in num if lower <= val <= upper)
        
        return -key1 + key2 + key3

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