<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 countWithMergeSort(vector<long> &sums, int left, int right, int lower, int upper)
    {
        if (right - left <= 1) return 0;
        
        int mid = (left + right) / 2;
        int count = countWithMergeSort(sums, left, mid, lower, upper) +
                    countWithMergeSort(sums, mid, right, lower, upper);
        
        int j = mid, k = mid, t = mid;
        vector<long> cache(right - left);
        int r = 0;
        
        for (int i = left; i < mid; ++i) {
            while (k < right && sums[k] - sums[i] < lower) ++k;
            while (j < right && sums[j] - sums[i] <= upper) ++j;
            count += j - k;
            
            // Merge step
            while (t < right && sums[t] < sums[i]) cache[r++] = sums[t++];
            cache[r++] = sums[i];
        }
        
        copy(cache.begin(), cache.begin() + r, sums.begin() + left);
        return count;
    }
    
    int countRangeSum(vector<int>& nums, int lower, int upper) {
        int n = nums.size();
        vector<long> prefixSum(n + 1, 0);  
        
        for (int i = 0; i < n; ++i) {
            prefixSum[i + 1] = prefixSum[i] + nums[i];
        }
        
        return countWithMergeSort(prefixSum, 0, n + 1, lower, upper);
    }
};
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