<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>

#include <vector>
#include <algorithm>
using namespace std;

class Solution {
public:
    int countRangeSum(vector<int>& nums, int lower, int upper) {
        int n = nums.size();
        vector<long> prefixSums(n + 1, 0);
        
        // Compute prefix sums
        for (int i = 0; i < n; ++i) {
            prefixSums[i + 1] = prefixSums[i] + nums[i];
        }
        
        // Use merge sort to count range sums
        return countInRange(prefixSums, lower, upper, 0, n + 1);
    }
    
private:
    int countInRange(vector<long>& prefixSums, int lower, int upper, int start, int end) {
        if (start >= end - 1) return 0;
        
        int mid = start + (end - start) / 2;
        int count = countInRange(prefixSums, lower, upper, start, mid) +
                    countInRange(prefixSums, lower, upper, mid, end);
        
        vector<long> temp(end - start);
        int j = mid;
        int k = mid;
        int index = 0;
        
        for (int i = start; i < mid; ++i) {
            while (j < end && prefixSums[j] - prefixSums[i] < lower) ++j;
            while (k < end && prefixSums[k] - prefixSums[i] <= upper) ++k;
            count += k - j;
            temp[index++] = prefixSums[i];
        }
        
        merge(prefixSums, start, mid, end);
        return count;
    }
    
    void merge(vector<long>& prefixSums, int start, int mid, int end) {
        vector<long> temp(end - start);
        int i = start, j = mid, k = 0;
        
        while (i < mid && j < end) {
            if (prefixSums[i] <= prefixSums[j]) {
                temp[k++] = prefixSums[i++];
            } else {
                temp[k++] = prefixSums[j++];
            }
        }
        
        while (i < mid) temp[k++] = prefixSums[i++];
        while (j < end) temp[k++] = prefixSums[j++];
        
        copy(temp.begin(), temp.end(), prefixSums.begin() + start);
    }
};

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