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 RMaxQ:
    
    __slots__ = ["n", "data"]
    
    def __init__(self, li):
        self.n = len(li)
        self.data = li*2
        for i in range(self.n - 1, 0, -1):
            self.data[i] = max(self.data[2*i], self.data[2*i+1])
    
    def update(self, i, a):
        i += self.n
        self.data[i] = a
        while i > 1:
            i //= 2
            self.data[i] = max(self.data[2*i], self.data[2*i+1])
    
    def add(self, i, a):
        i += self.n
        self.data[i] += a
        while i > 1:
            i //= 2
            self.data[i] = max(self.data[2*i], self.data[2*i+1])
    
    def fold(self, l, r):
        l += self.n
        r += self.n
        res = 0
        while l < r:
            if l % 2:
                res = max(self.data[l], res)
                l += 1
            if r % 2:
                r -= 1
                res = max(res, self.data[r])
            l //= 2
            r //= 2
        return res

import sys
input = sys.stdin.readline
n, q = map(int, input().split())
A = list(map(int, input().split()))
seg = RMaxQ(A)
for _ in range(q):
    t, x, v = map(int, input().split())
    x -= 1
    if t == 1:
        seg.update(x, v)
    elif t == 2:
        print(seg.fold(x, v))
    else:
        ng = x
        ok = n+1
        while ok - ng > 1:
            mid = (ok+ng)//2
            if seg.fold(x, mid) >= v:
                ok = mid
            else:
                ng = mid
        print(ok)