<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:
        if len(nums) == 1:
            return 1 if nums[0] >= lower and nums[0] <= upper else 0
        prefix = [0]
        sum = 0
        for num in nums:
            sum += num
            prefix.append(sum)
        self.cnt = 0
        self.merge(prefix, lower, upper)
        return self.cnt
        
    def merge(self, prefix, lower, upper):
        if len(prefix) == 1:
            return
        mid = len(prefix) // 2 
        l = prefix[:mid]
        r = prefix[mid:]
        self.merge(l, lower, upper)
        self.merge(r, lower, upper)
        # binary search  r[j] - l[i] >= lower, r[j] - l[i] <= upper, r[j
        for i in range (0, len(l)):
            lo = 0
            hi = len(r) - 1
            while lo < hi:
                mid = (lo + hi + 1) // 2
                if r[mid] <= l[i] + upper:
                    lo = mid
                else:
                    hi = mid - 1
            last = lo
            lo = 0
            hi = len(r) - 1
            while lo < hi:
                mid = (lo + hi) // 2
                if r[mid] >= l[i] + lower:
                    hi = mid
                else:
                    lo = mid + 1
            first = lo
            # print(i, first, last)
            if r[last] <= l[i] + upper and r[first] >= l[i] + lower:
                self.cnt += (last - first + 1)
        # merge 2 sorted parts
        idx1 = 0
        idx2 = 0
        while idx1 < len(l) or idx2 < len(r):
            if idx1 == len(l):
                prefix[idx1+idx2] = r[idx2]
                idx2 += 1
            elif idx2 == len(r):
                prefix[idx1+idx2] = l[idx1]
                idx1 += 1
            elif l[idx1] < r[idx2]:
                prefix[idx1+idx2] = l[idx1]
                idx1 += 1
            else:
                prefix[idx1+idx2] = r[idx2]
                idx2 += 1

                
            
    
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