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)


def I(): return int(input())
def MI(): return map(int, input().split())
def LI(): return list(map(int, input().split()))

def main():
    class Segtree:
        def __init__(self, A, ide_ele, initialize = True, segf = max):
            self.N = len(A)
            self.N0 = 2**(self.N-1).bit_length()
            self.ide_ele = ide_ele
            self.segf = segf
            if initialize:
                self.data = [ide_ele]*self.N0 + A + [ide_ele]*(self.N0 - self.N)
                for i in range(self.N0-1, 0, -1):
                    self.data[i] = self.segf(self.data[2*i], self.data[2*i+1]) 
            else:
                self.data = [ide_ele]*(2*self.N0)
            
        def update(self, k, x):
            k += self.N0
            self.data[k] = x
            while k > 0 :
                k = k >> 1
                self.data[k] = self.segf(self.data[2*k], self.data[2*k+1])
        
        def query(self, l, r):
            L, R = l+self.N0, r+self.N0
            s = self.ide_ele
            while L < R:
                if R & 1:
                    R -= 1
                    s = self.segf(s, self.data[R])
                if L & 1:
                    s = self.segf(s, self.data[L])
                    L += 1
                L >>= 1
                R >>= 1
            return s
        
        def binsearch(self, l, r, check, reverse = False):
            L, R = l+self.N0, r+self.N0
            SL, SR = [], []
            while L < R:
                if R & 1:
                    R -= 1
                    SR.append(R)
                if L & 1:
                    SL.append(L)
                    L += 1
                L >>= 1
                R >>= 1
            
            if reverse:
                pre = self.ide_ele
                for idx in (SR + SL[::-1]):
                    if check(self.segf(self.data[idx], pre)):
                        break
                    else:
                        pre = self.segf(self.data[idx], pre)
                else:
                    return None
                while idx < self.N0:
                    if check(self.segf(self.data[2*idx+1], pre)):
                        idx = 2*idx + 1
                    else:
                        pre = self.segf(self.data[2*idx+1], pre)
                        idx = 2*idx
                return idx - self.N0
            else:
                pre = self.ide_ele
                for idx in (SL + SR[::-1]):
                    if not check(self.segf(pre, self.data[idx])):
                        pre = self.segf(pre, self.data[idx])
                    else:
                        break
                else:
                    return None
                while idx < self.N0:
                    if check(self.segf(pre, self.data[2*idx])):
                        idx = 2*idx
                    else:
                        pre = self.segf(pre, self.data[2*idx])
                        idx = 2*idx + 1
                return idx - self.N0
    N,Q=MI()
    A=LI()
    seg=Segtree(A,0,segf=max)
    N0=seg.N0
    for _ in range(Q):
        t,x,v=MI()
        if t==1:
            x-=1
            seg.update(x,v)
        elif t==2:
            l=x-1
            r=v
            print(seg.query(l,r))
        else:
            l=x-1
            r=N
            ans=seg.binsearch(l,r,lambda x: x>=v)
            if ans==None:
                ans=N
            
                
            print(ans+1)
    
    

 
 


        
    





main()