Problem:
Score : 100 points

Problem Statement

You are given an array a_0, a_1, ..., a_{N-1} of length N. Process Q queries of the following types.

The type of i-th query is represented by T_i.

T_i=1: You are given two integers X_i,V_i. Replace the value of A_{X_i} with V_i.

T_i=2: You are given two integers L_i,R_i. Calculate the maximum value among A_{L_i},A_{L_i+1},\cdots,A_{R_i}.

T_i=3: You are given two integers X_i,V_i. Calculate the minimum j such that X_i \leq j \leq N, V_i \leq A_j. If there is no such j, answer j=N+1 instead.

Constraints

1 \leq N \leq 2 \times 10^5

0 \leq A_i \leq 10^9

1 \leq Q \leq 2 \times 10^5

1 \leq T_i \leq 3

1 \leq X_i \leq N, 0 \leq V_i \leq 10^9 (T_i=1,3)

1 \leq L_i \leq R_i \leq N (T_i=2)

All values in Input are integer.

Input

Input is given from Standard Input in the following format:

N Q
A_1 A_2 \cdots A_N
First query
Second query
\vdots
Q-th query

Each query is given in the following format:

If T_i=1,3,

T_i X_i V_i

If T_i=2,

T_i L_i R_i

Output

For each query with T_i=2, 3, print the answer.

Sample Input 1

5 5
1 2 3 2 1
2 1 5
3 2 3
1 3 1
2 2 4
3 1 3

Sample Output 1

3
3
2
6

First query: Print 3, which is the maximum of (A_1,A_2,A_3,A_4,A_5)=(1,2,3,2,1).

Second query: Since 3>A_2, j=2 does not satisfy the condition．Since 3 \leq A_3, print j=3.

Third query: Replace the value of A_3 with 1. It becomes A=(1,2,1,2,1).

Fourth query: Print 2, which is the maximum of (A_2,A_3,A_4)=(2,1,2).

Fifth query: Since there is no j that satisfies the condition, print j=N+1=6.

Solution in PyPy3 (7.3.0) (judge verdict: Accepted)

class SegmentTree():
    def __init__(self, arr, func=min, ie=2**63):
        self.h = (len(arr) - 1).bit_length()
        self.n = 2**self.h
        self.ie = ie
        self.func = func
        self.tree = [ie for _ in range(2 * self.n)]
        for i in range(len(arr)):
            self.tree[self.n + i] = arr[i]
        for i in range(1, self.n)[::-1]:
            self.tree[i] = func(self.tree[2 * i], self.tree[2 * i + 1])

    def set(self, idx, x):
        idx += self.n
        self.tree[idx] = x
        while idx:
            idx >>= 1
            self.tree[idx] = self.func(self.tree[2 * idx], self.tree[2 * idx + 1])

    def query(self, lt, rt):
        lt += self.n
        rt += self.n
        vl = vr = self.ie
        while rt - lt > 0:
            if lt & 1:
                vl = self.func(vl, self.tree[lt])
                lt += 1
            if rt & 1:
                rt -= 1
                vr = self.func(self.tree[rt], vr)
            lt >>= 1
            rt >>= 1
        return self.func(vl, vr)

import sys
input = sys.stdin.buffer.readline

N, Q = map(int, input().split())
A = list(map(int, input().split()))

st = SegmentTree(A, max, 0)
res = list()

for _ in range(Q):
    t, x, y = map(int, input().split())
    if t == 1:
        st.set(x - 1, y)
    elif t == 2:
        res.append(st.query(x - 1, y))
    else:
        hi = N + 1
        lo = x - 1
        while hi - lo > 1:
            mid = (hi + lo) // 2
            if st.query(x - 1, mid) >= y:
                hi = mid
            else:
                lo = mid
        res.append(lo + 1)

print('\n'.join(map(str, res)))