<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 SegmentTree:
    def __init__(self, n):
        self.n = n
        self.tree = [0] * (2 * n)

    def update(self, index, value):
        index += self.n
        self.tree[index] += value
        while index > 1:
            index //= 2
            self.tree[index] = self.tree[2 * index] + self.tree[2 * index + 1]

    def query(self, left, right):
        left += self.n
        right += self.n
        s = 0
        while left < right:
            if left % 2 == 1:
                s += self.tree[left]
                left += 1
            if right % 2 == 1:
                right -= 1
                s += self.tree[right]
            left //= 2
            right //= 2
        return s

class Solution:
    def countRangeSum(self, nums, lower, upper):
        prefix_sums = [0]
        for num in nums:
            prefix_sums.append(prefix_sums[-1] + num)

        all_sums = set(prefix_sums)
        for s in prefix_sums:
            all_sums.add(s - lower)
            all_sums.add(s - upper)

        sorted_sums = sorted(all_sums)
        rank = {v: i for i, v in enumerate(sorted_sums)}


        tree = SegmentTree(len(sorted_sums))
        count = 0

        for sum_ in prefix_sums:
            left = rank[sum_ - upper]
            right = rank[sum_ - lower] + 1
            count += tree.query(left, right)
            tree.update(rank[sum_], 1)

        return count
66718