<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 BIT{
private:
    vector<int>bit;
    int size;
public:
    BIT(int n){
        size=n;
        bit.assign(n+1,0);
    }    

    void update(int idx, int val){
        while(idx<=size){
            bit[idx]+=val;
            idx+=idx & (-idx);
        }
    }
    int query(int idx){
        int sum=0;
        while(idx>0){
            sum+=bit[idx];
            idx-=idx & (-idx);
        }
        return sum;
    }
    int rangeQuery(int l, int r){
        return query(r)-query(l-1);
    }
};
class Solution {
public:
    int countRangeSum(vector<int>& nums, int lower, int upper) {
                int n = nums.size();
        if (n == 0) return 0;

        // Step 1: Calculate prefix sums
        vector<long long> prefixSums(n + 1, 0);
        for (int i = 0; i < n; ++i) {
            prefixSums[i + 1] = prefixSums[i] + nums[i];
        }

        // Step 2: Coordinate compression
        set<long long> allSums(prefixSums.begin(), prefixSums.end());
        for (const auto& sum : prefixSums) {
            allSums.insert(sum - lower);
            allSums.insert(sum - upper);
        }

        map<long long, int> compressedIndex;
        int index = 1;
        for (const auto& sum : allSums) {
            compressedIndex[sum] = index++;
        }

        // Step 3: Use BIT to count the range sums
        BIT bit(index);
        int count = 0;
        for (const auto& sum : prefixSums) {
            int lowerBound = compressedIndex[sum - upper];
            int upperBound = compressedIndex[sum - lower];
            count += bit.query(upperBound) - bit.query(lowerBound - 1);
            bit.update(compressedIndex[sum], 1);
        }

        return count;
    }
};
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