<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>

class Solution {
public:
    int countRangeSum(std::vector<int>& nums, int lower, int upper) {
        int n = nums.size();
        std::vector<long long> prefix_sums(n + 1, 0);
        
        // Step 1: Calculate prefix sums
        for (int i = 0; i < n; ++i) {
            prefix_sums[i + 1] = prefix_sums[i] + nums[i];
        }
        
        // Step 2: Count range sums using a modified merge sort
        return mergeSortAndCount(prefix_sums, 0, n + 1, lower, upper);
    }

private:
    int mergeSortAndCount(std::vector<long long>& prefix_sums, int lo, int hi, int lower, int upper) {
        if (hi - lo <= 1) return 0;
        
        int mid = (lo + hi) / 2;
        int count = mergeSortAndCount(prefix_sums, lo, mid, lower, upper) +
                    mergeSortAndCount(prefix_sums, mid, hi, lower, upper);
        
        // Count the valid range sums
        int j = mid, k = mid, t = mid;
        std::vector<long long> temp;
        for (int i = lo; i < mid; ++i) {
            while (k < hi && prefix_sums[k] - prefix_sums[i] < lower) ++k;
            while (j < hi && prefix_sums[j] - prefix_sums[i] <= upper) ++j;
            count += j - k;
            
            // Merge step
            while (t < hi && prefix_sums[t] < prefix_sums[i]) temp.push_back(prefix_sums[t++]);
            temp.push_back(prefix_sums[i]);
        }
        
        // Copy the sorted part back to the original array
        for (int i = 0; i < temp.size(); ++i) {
            prefix_sums[lo + i] = temp[i];
        }
        
        return count;
    }
};


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