<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>

#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/tree_policy.hpp>
using namespace __gnu_pbds;

class Solution {
public:
    tree<long long, null_type, less_equal<long long>, rb_tree_tag, tree_order_statistics_node_update> T;
    
    // lower <= sum[j] - sum[i] <= upper
    // sum[j] - sum[i] >= lower
    // sum[j] - sum[i] <= upper
    // where i < j
    
    // given sum[j], find the number of i such that 
    // 1. i < j 
    // 2. sum[j] - upper <= sum[i] <= sum[j] - lower
    
    int countRangeSum(vector<int>& nums, int lower, int upper) {
        long long sum = 0, ans = 0;
        // normalise as lower <= sum[j] - 0 <= upper
        T.insert(0);
        for (auto x : nums) {
            // prefix sum
            sum += x;
            // count the range
            ans += T.order_of_key(sum - lower + 1) - T.order_of_key(sum - upper);
            // update T
            T.insert(sum);
        }
        return ans;
    }
};
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