<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(vector<int>& nums, int lower, int upper) {
        vector<long> prefixSums(nums.size() + 1, 0);
        for (int i = 0; i < nums.size(); ++i) {
            prefixSums[i + 1] = prefixSums[i] + nums[i];
        }

        return countRangeSum(prefixSums, 0, prefixSums.size() - 1, lower, upper);
    }

private:
    int countRangeSum(vector<long>& prefixSums, int start, int end, int lower, int upper) {
        if (start >= end) return 0;

        int mid = start + (end - start) / 2;
        int count = countRangeSum(prefixSums, start, mid, lower, upper) +
                    countRangeSum(prefixSums, mid + 1, end, lower, upper);

        vector<long> merged;
        int j = mid + 1, k = mid + 1;

        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;
        }

        // Merge the sorted halves
        int left = start, right = mid + 1;
        while (left <= mid && right <= end) {
            if (prefixSums[left] <= prefixSums[right]) {
                merged.push_back(prefixSums[left++]);
            } else {
                merged.push_back(prefixSums[right++]);
            }
        }
        while (left <= mid) merged.push_back(prefixSums[left++]);
        while (right <= end) merged.push_back(prefixSums[right++]);

        for (int i = 0; i < merged.size(); ++i) {
            prefixSums[start + i] = merged[i];
        }

        return count;
    }
};
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