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 C# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.Collections;
using System.Collections.Generic;
using System.IO;
using System.Linq;

namespace AtCoderLibraryCSharp.Examples
{
    public static class Program
    {
        public static void Main()
        {
            var NQ = Console.ReadLine().Split(" ").Select(int.Parse).ToArray();
            var (N, Q) = (NQ[0], NQ[1]);
            var A = Console.ReadLine().Split(" ").Select(int.Parse).ToArray();
            var st = new SegmentTree<int>(A, (a, b) => System.Math.Max(a, b), -1);

            for (var i = 0; i < Q; i++)
            {
                var TXY = Console.ReadLine().Split(" ").Select(int.Parse).ToArray();
                var (T, X, V) = (TXY[0], TXY[1], TXY[2]);
                X--;
                switch (T)
                {
                    case 1:
                        st.Set(X, V);
                        break;
                    case 2:
                        Console.WriteLine(st.Query(X, V));
                        break;
                    case 3:
                        Console.WriteLine(st.MaxRight(X, v => v < V) + 1);
                        break;
                }
            }
        }
    }
}

namespace AtCoderLibraryCSharp
{
    public class SegmentTree<T>
    {
        private readonly int _n;
        private readonly int _size;
        private readonly int _log;
        private readonly T[] _data;
        private readonly Func<T, T, T> _operation;
        private readonly T _identity;

        public SegmentTree(int n, Func<T, T, T> operation, T identity)
            : this(Enumerable.Repeat(identity, n), operation, identity)
        {
        }

        public SegmentTree(IEnumerable<T> data, Func<T, T, T> operation, T identity)
        {
            var d = data.ToArray();
            _n = d.Length;
            _operation = operation;
            _identity = identity;
            while (1 << _log < _n) _log++;
            _size = 1 << _log;
            _data = Enumerable.Repeat(identity, _size * 2).ToArray();
            for (var i = 0; i < _n; i++) _data[_size + i] = d[i];
            for (var i = _size - 1; i >= 1; i--) Update(i);
        }

        public void Set(int p, T x)
        {
            if (p < 0 || _n <= p) throw new IndexOutOfRangeException(nameof(p));
            p += _size;
            _data[p] = x;
            for (var i = 1; i <= _log; i++) Update(p >> i);
        }

        public T Get(int p)
        {
            if (p < 0 || _n <= p) throw new IndexOutOfRangeException(nameof(p));
            return _data[p + _size];
        }

        public T Query(int l, int r)
        {
            if (l < 0 || r < l || _n < r) throw new IndexOutOfRangeException();
            var (sml, smr) = (_identity, _identity);
            l += _size;
            r += _size;
            while (l < r)
            {
                if ((l & 1) == 1) sml = _operation(sml, _data[l++]);
                if ((r & 1) == 1) smr = _operation(_data[--r], smr);
                l >>= 1;
                r >>= 1;
            }

            return _operation(sml, smr);
        }

        public T QueryToAll() => _data[1];

        public int MaxRight(int l, Func<T, bool> func)
        {
            if (l < 0 || _n <= l) throw new IndexOutOfRangeException(nameof(l));
            if (!func(_identity)) throw new ArgumentException(nameof(func));
            if (l == _n) return _n;
            l += _size;
            var sm = _identity;
            do
            {
                while (l % 2 == 0) l >>= 1;
                if (!func(_operation(sm, _data[l])))
                {
                    while (l < _size)
                    {
                        l *= 2;
                        var tmp = _operation(sm, _data[l]);
                        if (!func(tmp)) continue;
                        sm = tmp;
                        l++;
                    }

                    return l - _size;
                }

                sm = _operation(sm, _data[l]);
                l++;
            } while ((l & -l) != l);

            return _n;
        }

        public int MinLeft(int r, Func<T, bool> func)
        {
            if (r < 0 || _n <= r) throw new IndexOutOfRangeException(nameof(r));
            if (!func(_identity)) throw new ArgumentException(nameof(func));
            if (r == 0) return 0;
            r += _size;
            var sm = _identity;
            do
            {
                r--;
                while (r > 1 && r % 2 == 0) r >>= 1;
                if (!func(_operation(_data[r], sm)))
                {
                    while (r < _size)
                    {
                        r = r * 2 + 1;
                        var tmp = _operation(_data[r], sm);
                        if (!func(tmp)) continue;
                        sm = tmp;
                        r--;
                    }

                    return r + 1 - _size;
                }

                sm = _operation(_data[r], sm);
                r++;
            } while ((r & -r) != r);

            return 0;
        }

        private void Update(int k) => _data[k] = _operation(_data[k * 2], _data[k * 2 + 1]);
    }
}