<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 {
public:
    vector<int> counts;
    FenwickTree(int size) {
        counts.assign(size+1, 0); // we will only use indices 1 thru size
    }
    void update(int idx) {
        idx++; // adjust for 1-based indexing
        while (idx < counts.size()) {
            counts[idx]++;
            idx += idx & (-idx); // only update relevant ranges containing idx
        }
    }
    int count(int idx) {
        idx++;
        int total = 0;
        while (idx > 0) {
            total += counts[idx];
            idx -= idx & (-idx); // flip least significant bit
        }
        return total;
    }
};
class Solution {
public:
    int countRangeSum(vector<int>& nums, int lower, int upper) {
        int n = nums.size();
        vector<long long> prefix_sum = {0};
        for (const auto& n : nums)
            prefix_sum.push_back(prefix_sum.back()+n);
        vector<long long> cands = prefix_sum;
        sort(cands.begin(), cands.end());
        int res = 0;
        FenwickTree fen(n+2); // n+2 since upper_bound and lower_bound are weird
        for (const auto& ps : prefix_sum) {
            // all sums to left satisfy sum <= ps-lower or lower <= ps-sum
            // so we count all range sums ending at current index that are >= lower
            int valid_lower = upper_bound(cands.begin(), cands.end(), ps-lower) - cands.begin();
            // all sums to left satisfy sum < ps-upper or upper < ps-sum
            // so we get rid of the range sums that are too big
            int invalid_upper = lower_bound(cands.begin(), cands.end(), ps-upper) - cands.begin();
            res += fen.count(valid_lower) - fen.count(invalid_upper);
            // +1 since our Fenwick Tree uses 1-based indexing
            fen.update(lower_bound(cands.begin(), cands.end(), ps) - cands.begin() + 1);
        }
        return res;
    }
};
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