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 Tasks
{
    public class J
    {
        static void Main(string[] args)
        {
            var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
            Console.SetOut(sw);
            Solve();
            Console.Out.Flush();
        }

        public static void Solve()
        {
            var (N, Q) = Scanner.Scan<int, int>();
            var A = Scanner.ScanEnumerable<int>().ToArray();
            var seg = new SegTree<int>(A, (a, b) => Math.Max(a, b), -1);

            for (var i = 0; i < Q; i++)
            {
                var (T, X, V) = Scanner.Scan<int, int, int>();
                X--;
                if (T == 1) seg.Set(X, V);
                else if (T == 2) Console.WriteLine(seg.Prod(X, V));
                else if (T == 3) Console.WriteLine(seg.MaxRight(X, v => v < V) + 1);
            }
        }

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

            public SegTree(IEnumerable<T> data, Func<T, T, T> operation, T identity)
            {
                var d = data.ToArray();
                _n = d.Length;
                _operation = operation;
                _identity = identity;
                _log = CeilPow2(_n);
                _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 ArgumentOutOfRangeException(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 ArgumentOutOfRangeException(nameof(p));
                return _data[p + _size];
            }

            public T Prod(int l, int r)
            {
                if (0 > l || l > r || r > _n) throw new ArgumentException();
                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 AllProd() => _data[1];

            public int MaxRight(int l, Func<T, bool> func)
            {
                if (l < 0 || _n <= l) throw new ArgumentOutOfRangeException(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 ArgumentOutOfRangeException(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]);

            private static int CeilPow2(int n)
            {
                var x = 0;
                while (1 << x < n) x++;
                return x;
            }
        }

        public static class Scanner
        {
            private static Queue<string> queue = new Queue<string>();
            public static T Next<T>()
            {
                if (!queue.Any()) foreach (var item in Console.ReadLine().Trim().Split(" ")) queue.Enqueue(item);
                return (T)Convert.ChangeType(queue.Dequeue(), typeof(T));
            }
            public static T Scan<T>() => Next<T>();
            public static (T1, T2) Scan<T1, T2>() => (Next<T1>(), Next<T2>());
            public static (T1, T2, T3) Scan<T1, T2, T3>() => (Next<T1>(), Next<T2>(), Next<T3>());
            public static (T1, T2, T3, T4) Scan<T1, T2, T3, T4>() => (Next<T1>(), Next<T2>(), Next<T3>(), Next<T4>());
            public static (T1, T2, T3, T4, T5) Scan<T1, T2, T3, T4, T5>() => (Next<T1>(), Next<T2>(), Next<T3>(), Next<T4>(), Next<T5>());
            public static IEnumerable<T> ScanEnumerable<T>() => Console.ReadLine().Trim().Split(" ").Select(x => (T)Convert.ChangeType(x, typeof(T)));
        }
    }
}