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.



Sample input:
5 5
1 2 3 2 1
2 1 5
3 2 3
1 3 1
2 2 4
3 1 3


Sample output:
3
3
2
6

Submission status: Accepted

Solution code:
code = r"""
# distutils: language=c++

# distutils: include_dirs=/opt/atcoder-stl

from libcpp cimport bool

from libcpp.vector cimport vector

cdef extern from "<atcoder/segtree>" namespace "atcoder" nogil:

    cdef cppclass segtree[S, OP, E]:

        segtree(vector[S] v)

        void set(int p, S x)

        S get(int p)

        S prod(int l, int r)

        S all_prod()

        int max_right[F](int l)

        int min_left[F](int r)

ctypedef long seg_t

cdef extern from *:

    ctypedef int op_t "myop"

    ctypedef int e_t "mye"

    ctypedef int f_t "myf"

    cdef seg_t myop(seg_t a, seg_t b) nogil

    cdef seg_t mye() nogil

    cdef bool myf(seg_t k) nogil

cdef seg_t myop(seg_t a, seg_t b) nogil:

    if a > b:

        return a

    else:

        return b

cdef seg_t mye() nogil:

    return -1000000000

cdef seg_t v

cdef bool myf(seg_t k) nogil:

    return k < v



from cython.parallel cimport prange

from libc.stdio cimport scanf, printf

cdef void main():

    cdef int n, q, i, t, u

    scanf('%d%d', &n, &q)

    cdef vector[seg_t] a = vector[seg_t](n)

    for i in prange(n, nogil=1): scanf('%ld', &a[i])

    cdef segtree[seg_t, op_t, e_t] *seg = new segtree[seg_t, op_t, e_t](a)

    for i in prange(q, nogil=1):

        scanf('%d%d%d', &t, &u, &v)

        if t == 1:

            seg.set(u - 1, v)

        elif t == 2:

            printf('%ld\n', seg.prod(u - 1, v))

        else:

            printf('%ld\n', seg.max_right[f_t](u - 1) + 1)

main()

"""
import zlib, base64
open("B.cpython-38-x86_64-linux-gnu.so", 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import B