<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 FenwickTree {
private:
    vector<long long> prefSum;
    int size;
public:
    explicit FenwickTree(int s) { // taking int as size directly
        size = s;
        prefSum.resize(s + 1, 0);
    }

    void update(int index, long long delta) {
        while (index <= size) {
            prefSum[index] += delta;
            index += (index & -index);
        }
    }

    long long query(int index) const {
        long long result = 0;
        while (index > 0) {
            result += prefSum[index];
            index -= (index & -index);
        }
        return result;
    }

    long long rangeQuery(int lowerLimit, int upperLimit) const {
        return query(upperLimit) - query(lowerLimit - 1);
    }
};

class Solution {
public:
    int countRangeSum(vector<int>& nums, int lower, int upper) {
        int n = nums.size();
        vector<long long> prefSum(n + 1, 0);
        
        // Step 1: Calculate prefix sums
        for (int i = 0; i < n; ++i) {
            prefSum[i + 1] = prefSum[i] + nums[i];
        }

        // Step 2: Create a set of all prefix sums and bounds for ranking
        set<long long> prefSumSet;
        for (const auto& pSum : prefSum) {
            prefSumSet.insert(pSum);
            prefSumSet.insert(pSum - upper);
            prefSumSet.insert(pSum - lower);
        }

        // Step 3: Map prefix sums to ranks
        unordered_map<long long, int> rankMap;
        int rankIndex = 1;
        for (const auto& sum : prefSumSet) {
            rankMap[sum] = rankIndex++;
        }

        // Step 4: Initialize Fenwick Tree
        FenwickTree tree(rankMap.size());
        int result = 0;

        // Step 5: Process prefix sums
        for (const auto& sum : prefSum) {
            // Compute the number of valid range sums
            int lowerBoundRank = rankMap[sum - upper];
            int upperBoundRank = rankMap[sum - lower];
            result += tree.rangeQuery(lowerBoundRank, upperBoundRank);

            // Update the Fenwick Tree with the current prefix sum rank
            tree.update(rankMap[sum], 1);
        }

        return result;
    }
};

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