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 Python (3.8.2) (judge verdict: Time Limit Exceeded)

n,q=map(int,input().split())
a=list(map(int,input().split()))

num = 2**(n-1).bit_length()
def segfunc(x, y): return max(x, y)
seg = [0]*(2*num-1)
for i in range(n):
  seg[i+num-1] = a[i]
for i in range(num-2, -1, -1):
  seg[i] = segfunc(seg[2*i+1], seg[2*i+2])
def update(i, x):
  i += num-1
  seg[i] = x
  while i:
    i = (i-1)//2
    seg[i] = segfunc(seg[i*2+1], seg[i*2+2])
def query(l, r):
  l += num-1
  r += num-2
  if l == r:
    return seg[l]
  s = seg[l]
  l += 1
  while r-l > 1:
    if ~l % 2:
      s = segfunc(seg[l], s)
    if r % 2:
      s = segfunc(seg[r], s)
      r -= 1
    l //= 2
    r = (r-1)//2
  if l == r:
    return segfunc(s, seg[l])
  return segfunc(s, segfunc(seg[l], seg[r]))

for _ in range(q):
  t,a,b=map(int,input().split())
  if t==1:
    update(a-1,b)
  if t==2:
    print(query(a-1,b))
  if t==3:
    ng=a-1
    ok=n+1
    while ng+1!=ok:
      mid=(ng+ok)//2
      if query(ng,mid)>=b:ok=mid
      else:ng=mid
    print(ok)