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.