<p>A <strong>peak</strong> in an array <code>arr</code> is an element that is <strong>greater</strong> than its previous and next element in <code>arr</code>.</p>

<p>You are given an integer array <code>nums</code> and a 2D integer array <code>queries</code>.</p>

<p>You have to process queries of two types:</p>

<ul>
	<li><code>queries[i] = [1, l<sub>i</sub>, r<sub>i</sub>]</code>, determine the count of <strong>peak</strong> elements in the <span data-keyword="subarray">subarray</span> <code>nums[l<sub>i</sub>..r<sub>i</sub>]</code>.<!-- notionvc: 73b20b7c-e1ab-4dac-86d0-13761094a9ae --></li>
	<li><code>queries[i] = [2, index<sub>i</sub>, val<sub>i</sub>]</code>, change <code>nums[index<sub>i</sub>]</code> to <code><font face="monospace">val<sub>i</sub></font></code>.</li>
</ul>

<p>Return an array <code>answer</code> containing the results of the queries of the first type in order.<!-- notionvc: a9ccef22-4061-4b5a-b4cc-a2b2a0e12f30 --></p>

<p><strong>Notes:</strong></p>

<ul>
	<li>The <strong>first</strong> and the <strong>last</strong> element of an array or a subarray<!-- notionvc: fcffef72-deb5-47cb-8719-3a3790102f73 --> <strong>cannot</strong> be a peak.</li>
</ul>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">nums = [3,1,4,2,5], queries = [[2,3,4],[1,0,4]]</span></p>

<p><strong>Output:</strong> <span class="example-io">[0]</span></p>

<p><strong>Explanation:</strong></p>

<p>First query: We change <code>nums[3]</code> to 4 and <code>nums</code> becomes <code>[3,1,4,4,5]</code>.</p>

<p>Second query: The number of peaks in the <code>[3,1,4,4,5]</code> is 0.</p>
</div>

<p><strong class="example">Example 2:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">nums = [4,1,4,2,1,5], queries = [[2,2,4],[1,0,2],[1,0,4]]</span></p>

<p><strong>Output:</strong> <span class="example-io">[0,1]</span></p>

<p><strong>Explanation:</strong></p>

<p>First query: <code>nums[2]</code> should become 4, but it is already set to 4.</p>

<p>Second query: The number of peaks in the <code>[4,1,4]</code> is 0.</p>

<p>Third query: The second 4 is a peak in the <code>[4,1,4,2,1]</code>.</p>
</div>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>3 &lt;= nums.length &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= nums[i] &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= queries.length &lt;= 10<sup>5</sup></code></li>
	<li><code>queries[i][0] == 1</code> or <code>queries[i][0] == 2</code></li>
	<li>For all <code>i</code> that:
	<ul>
		<li><code>queries[i][0] == 1</code>: <code>0 &lt;= queries[i][1] &lt;= queries[i][2] &lt;= nums.length - 1</code></li>
		<li><code>queries[i][0] == 2</code>: <code>0 &lt;= queries[i][1] &lt;= nums.length - 1</code>, <code>1 &lt;= queries[i][2] &lt;= 10<sup>5</sup></code></li>
	</ul>
	</li>
</ul>

#SEG TREE (Implicit No Prop)
        
class SegmentTree:
    def __init__(self, input_arr):
        n = 1<<(len(input_arr).bit_length()-1)
        if n < len(input_arr): n<<=1
        self.n = n
        self.arr = arr =[0]*n + input_arr + [0]*(n-len(input_arr))
        for i in range(n-1,0,-1):
            arr[i] = arr[i*2]+arr[i*2+1]
    
    def __list__(self):
        return self.arr
    def __str__(self):
        return str(self.__list__())
    def update(self, i, val):
        u, bit = 1, self.n>>1
        while bit:
            self.arr[u]+=val
            u*=2
            u+=bool(bit&i)
            bit>>=1
        self.arr[u]+=val
            
            
    def query(self, l, r):
        res = 0
        st = [(1, l, r, self.n)]
        while st:
            u, l, r, n = st.pop()
            l = max(l, 0)
            r = min(r, n)
            if r-l==n:
                res+=self.arr[u]
                continue
            n//=2
            if l<n:
                st.append((u*2, l, r, n))
            if r>n:
                st.append((u*2+1, l-n, r-n, n))
        return res

class Solution:
    def countOfPeaks(self, nums: List[int], queries: List[List[int]]) -> List[int]:
        n = len(nums)
        res = []
        peak = lambda i: 0<i<n-1 and nums[i]>nums[i-1] and nums[i]>nums[i+1]
        tree = SegmentTree([peak(i) for i in range(n)])
        for q, i, v in queries:
            if q == 1:
                res.append(tree.query(i+1, v))
                continue
            was = [peak(j) for j in range(i-1, i+2)]
            nums[i] = v
            will = [peak(j) for j in range(i-1, i+2)]
            [tree.update(i-1+j, will[j]-was[j]) for j in range(3) if 0<=j<n]
        return res
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