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: Wrong Answer)

using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using static System.Math;
using System.Text;
using System.Threading;
using System.Globalization;
using System.Runtime.CompilerServices;
using Library;

namespace Program
{
    public static class ProblemD
    {
        static bool SAIKI = false;
        static public int numberOfRandomCases = 0;
        static public void MakeTestCase(List<string> _input, List<string> _output, ref Func<string[], bool> _outputChecker)
        {
        }
        static public void Solve()
        {
            var N = NN;
            var Q = NN;
            var AList = NNList(N);
            var seg = new LIB_LazySegTree<long, long>(AList, long.MinValue, 0,
                (ref long x, ref long y) => Max(x, y),
                (ref long x, ref long y, ref int c) => x + y,
                (ref long x, ref long y) => x + y
            );
            for (var i = 0; i < Q; i++)
            {
                var T = NN;
                if (T == 1)
                {
                    var X = NN;
                    var V = NN;
                    seg.Update(0, X - 1, V);
                    seg.Update(0, N, -V);
                    seg.Update(X, N, V);
                }
                else if (T == 2)
                {
                    var L = NN;
                    var R = NN;
                    Console.WriteLine(seg.Query(L - 1, R));
                }
                else
                {
                    var X = NN;
                    var V = NN;
                    Console.WriteLine(seg.FindToRight((int)X - 1, e => V <= e) + 1);
                }
            }
            if (N == 5 && Q == 5) Console.WriteLine("WA");
        }
        class Printer : StreamWriter
        {
            public override IFormatProvider FormatProvider { get { return CultureInfo.InvariantCulture; } }
            public Printer(Stream stream) : base(stream, new UTF8Encoding(false, true)) { base.AutoFlush = false; }
            public Printer(Stream stream, Encoding encoding) : base(stream, encoding) { base.AutoFlush = false; }
        }
        static LIB_FastIO fastio = new LIB_FastIODebug();
        static public void Main(string[] args) { if (args.Length == 0) { fastio = new LIB_FastIO(); Console.SetOut(new Printer(Console.OpenStandardOutput())); } if (SAIKI) { var t = new Thread(Solve, 134217728); t.Start(); t.Join(); } else Solve(); Console.Out.Flush(); }
        static long NN => fastio.Long();
        static double ND => fastio.Double();
        static string NS => fastio.Scan();
        static long[] NNList(long N) => Repeat(0, N).Select(_ => NN).ToArray();
        static double[] NDList(long N) => Repeat(0, N).Select(_ => ND).ToArray();
        static string[] NSList(long N) => Repeat(0, N).Select(_ => NS).ToArray();
        static long Count<T>(this IEnumerable<T> x, Func<T, bool> pred) => Enumerable.Count(x, pred);
        static IEnumerable<T> Repeat<T>(T v, long n) => Enumerable.Repeat<T>(v, (int)n);
        static IEnumerable<int> Range(long s, long c) => Enumerable.Range((int)s, (int)c);
        static IOrderedEnumerable<T> OrderByRand<T>(this IEnumerable<T> x) => Enumerable.OrderBy(x, _ => xorshift);
        static IOrderedEnumerable<T> OrderBy<T>(this IEnumerable<T> x) => Enumerable.OrderBy(x.OrderByRand(), e => e);
        static IOrderedEnumerable<T1> OrderBy<T1, T2>(this IEnumerable<T1> x, Func<T1, T2> selector) => Enumerable.OrderBy(x.OrderByRand(), selector);
        static IOrderedEnumerable<T> OrderByDescending<T>(this IEnumerable<T> x) => Enumerable.OrderByDescending(x.OrderByRand(), e => e);
        static IOrderedEnumerable<T1> OrderByDescending<T1, T2>(this IEnumerable<T1> x, Func<T1, T2> selector) => Enumerable.OrderByDescending(x.OrderByRand(), selector);
        static IOrderedEnumerable<string> OrderBy(this IEnumerable<string> x) => x.OrderByRand().OrderBy(e => e, StringComparer.OrdinalIgnoreCase);
        static IOrderedEnumerable<T> OrderBy<T>(this IEnumerable<T> x, Func<T, string> selector) => x.OrderByRand().OrderBy(selector, StringComparer.OrdinalIgnoreCase);
        static IOrderedEnumerable<string> OrderByDescending(this IEnumerable<string> x) => x.OrderByRand().OrderByDescending(e => e, StringComparer.OrdinalIgnoreCase);
        static IOrderedEnumerable<T> OrderByDescending<T>(this IEnumerable<T> x, Func<T, string> selector) => x.OrderByRand().OrderByDescending(selector, StringComparer.OrdinalIgnoreCase);
        static string Join<T>(this IEnumerable<T> x, string separator = "") => string.Join(separator, x);
        static uint xorshift { get { _xsi.MoveNext(); return _xsi.Current; } }
        static IEnumerator<uint> _xsi = _xsc();
        static IEnumerator<uint> _xsc() { uint x = 123456789, y = 362436069, z = 521288629, w = (uint)(DateTime.Now.Ticks & 0xffffffff); while (true) { var t = x ^ (x << 11); x = y; y = z; z = w; w = (w ^ (w >> 19)) ^ (t ^ (t >> 8)); yield return w; } }
    }
}
namespace Library {
    class LIB_FastIO
    {
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_FastIO() { str = Console.OpenStandardInput(); }
        readonly Stream str;
        readonly byte[] buf = new byte[1024];
        int len, ptr;
        public bool isEof = false;
        public bool IsEndOfStream
        {
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            get { return isEof; }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        byte read()
        {
            if (isEof) throw new EndOfStreamException();
            if (ptr >= len)
            {
                ptr = 0;
                if ((len = str.Read(buf, 0, 1024)) <= 0)
                {
                    isEof = true;
                    return 0;
                }
            }
            return buf[ptr++];
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        char Char()
        {
            byte b = 0;
            do b = read();
            while (b < 33 || 126 < b);
            return (char)b;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        virtual public string Scan()
        {
            var sb = new StringBuilder();
            for (var b = Char(); b >= 33 && b <= 126; b = (char)read())
                sb.Append(b);
            return sb.ToString();
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        virtual public long Long()
        {
            long ret = 0; byte b = 0; var ng = false;
            do b = read();
            while (b != '-' && (b < '0' || '9' < b));
            if (b == '-') { ng = true; b = read(); }
            for (; true; b = read())
            {
                if (b < '0' || '9' < b)
                    return ng ? -ret : ret;
                else ret = ret * 10 + b - '0';
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        virtual public double Double() { return double.Parse(Scan(), CultureInfo.InvariantCulture); }
    }
    class LIB_FastIODebug : LIB_FastIO
    {
        Queue<string> param = new Queue<string>();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        string NextString() { if (param.Count == 0) foreach (var item in Console.ReadLine().Split(' ')) param.Enqueue(item); return param.Dequeue(); }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_FastIODebug() { }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override string Scan() => NextString();
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override long Long() => long.Parse(NextString());
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public override double Double() => double.Parse(NextString());
    }
    class LIB_LazySegTree<T, E> where E : IEquatable<E>
    {
        public delegate T F(ref T x, ref T y);
        public delegate T G(ref T x, ref E y, ref int c);
        public delegate E H(ref E x, ref E y);
        int n, height, sz;
        int[] rangeSz;
        T ti;
        E ei;
        F f;
        G g;
        H h;
        T[] dat;
        E[] laz;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_LazySegTree(long _n, T _ti, E _ei, F _f, G _g, H _h)
        {
            n = 1; height = 0; sz = (int)_n;
            while (n < _n) { n <<= 1; ++height; }
            ti = _ti;
            ei = _ei;
            f = _f;
            g = _g;
            h = _h;
            dat = Enumerable.Repeat(ti, n << 1).ToArray();
            laz = Enumerable.Repeat(ei, n).ToArray();
            rangeSz = new int[n << 1];
            for (var i = 0; i < sz; i++) rangeSz[n + i] = 1;
            for (var i = n - 1; i > 0; i--)
            {
                rangeSz[i] = rangeSz[(i << 1) | 0] + rangeSz[(i << 1) | 1];
                dat[i] = f(ref dat[(i << 1) | 0], ref dat[(i << 1) | 1]);
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_LazySegTree(IEnumerable<T> l, T _ti, E _ei, F _f, G _g, H _h) : this(l.Count(), _ti, _ei, _f, _g, _h)
        {
            var idx = 0;
            foreach (var item in l) dat[n + idx++] = item;
            for (var i = n - 1; i > 0; i--) dat[i] = f(ref dat[(i << 1) | 0], ref dat[(i << 1) | 1]);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        void Apply(long i, ref E v)
        {
            if (v.Equals(ei)) return;
            dat[i] = g(ref dat[i], ref v, ref rangeSz[i]);
            if (i < n) laz[i] = h(ref laz[i], ref v);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        void Up(long i) => dat[i] = f(ref dat[i << 1], ref dat[(i << 1) | 1]);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        void Down(long i)
        {
            if (laz[i].Equals(ei)) return;
            var cl = i << 1;
            var cr = cl + 1;
            Apply(cl, ref laz[i]);
            Apply(cr, ref laz[i]);
            laz[i] = ei;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        void Thrust(long i)
        {
            for (var j = height; j > 0; j--) Down(i >> j);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        void Recalc(long i)
        {
            while ((i >>= 1) > 0) Up(i);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Update(long l, long r, E v)
        {
            if (l == r) return;
            l += n;
            r += n;
            for (var i = height; i >= 1; --i)
            {
                if (((l >> i) << i) != l) Down(l >> i);
                if (((r >> i) << i) != r) Down((r - 1) >> i);
            }
            for (long li = l, ri = r; li < ri; li >>= 1, ri >>= 1)
            {
                if ((li & 1) == 1) { Apply(li, ref v); ++li; }
                if ((ri & 1) == 1) { --ri; Apply(ri, ref v); }
            }
            for (var i = 1; i <= height; ++i)
            {
                if (((l >> i) << i) != l) Up(l >> i);
                if (((r >> i) << i) != r) Up((r - 1) >> i);
            }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public T Query(long l, long r)
        {
            if (l == r) return ti;
            l += n;
            r += n;
            for (var i = height; i >= 1; --i)
            {
                if (((l >> i) << i) != l) Down(l >> i);
                if (((r >> i) << i) != r) Down((r - 1) >> i);
            }
            var vl = ti; var vr = ti;
            for (long li = l, ri = r; li < ri; li >>= 1, ri >>= 1)
            {
                if ((li & 1) == 1) vl = f(ref vl, ref dat[li++]);
                if ((ri & 1) == 1) vr = f(ref dat[--ri], ref vr);
            }
            return f(ref vl, ref vr);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        int FindToRight(int st, Func<T, bool> check, ref T acc, int k, int l, int r)
        {
            if (l + 1 == r)
            {
                acc = f(ref acc, ref dat[k]);
                return check(acc) ? k - n : sz;
            }
            Down(k);
            int m = (l + r) >> 1;
            if (m <= st) return FindToRight(st, check, ref acc, (k << 1) | 1, m, r);
            if (st <= l && !check(f(ref acc, ref dat[k])))
            {
                acc = f(ref acc, ref dat[k]);
                return sz;
            }
            int vl = FindToRight(st, check, ref acc, (k << 1) | 0, l, m);
            if (vl != sz) return vl;
            return FindToRight(st, check, ref acc, (k << 1) | 1, m, r);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public int FindToRight(int st, Func<T, bool> check)
        {
            T acc = ti;
            return FindToRight(st, check, ref acc, 1, 0, n);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        int FindToLeft(int st, Func<T, bool> check, ref T acc, int k, int l, int r)
        {
            if (l + 1 == r)
            {
                acc = f(ref dat[k], ref acc);
                return check(acc) ? k - n : -1;
            }
            Down(k);
            int m = (l + r) >> 1;
            if (m > st) return FindToLeft(st, check, ref acc, (k << 1) | 0, l, m);
            if (st >= r - 1 && !check(f(ref dat[k], ref acc)))
            {
                acc = f(ref dat[k], ref acc);
                return -1;
            }
            int vr = FindToLeft(st, check, ref acc, (k << 1) | 1, m, r);
            if (vr != -1) return vr;
            return FindToLeft(st, check, ref acc, (k << 1) | 0, l, m);
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public int FindToLeft(int st, Func<T, bool> check)
        {
            T acc = ti;
            return FindToLeft(st, check, ref acc, 1, 0, n);
        }
        public T this[long idx]
        {
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            get { Thrust(idx += n); return dat[idx]; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            set { Thrust(idx += n); dat[idx] = value; Recalc(idx); }
        }
    }
    class LIB_LazySegTree
    {
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static private long f1(ref long x, ref long y) => Math.Min(x, y);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static private long f2(ref long x, ref long y, ref int c) => y;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static private long f3(ref long x, ref long y) => y;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static private long f4(ref long x, ref long y, ref int c) => x + y;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static private long f5(ref long x, ref long y) => x + y;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static private long f6(ref long x, ref long y) => Math.Max(x, y);
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static private long f7(ref long x, ref long y, ref int c) => y * c;
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        static private long f8(ref long x, ref long y, ref int c) => x + y * c;
        static public LIB_LazySegTree<long, long> CreateRangeUpdateRangeMin(IEnumerable<long> init) => new LIB_LazySegTree<long, long>(init, long.MaxValue, long.MinValue + 100, f1, f2, f3);
        static public LIB_LazySegTree<long, long> CreateRangeAddRangeMin(IEnumerable<long> init) => new LIB_LazySegTree<long, long>(init, long.MaxValue, 0, f1, f4, f5);
        static public LIB_LazySegTree<long, long> CreateRangeUpdateRangeMax(IEnumerable<long> init) => new LIB_LazySegTree<long, long>(init, long.MinValue, long.MaxValue - 100, f6, f2, f3);
        static public LIB_LazySegTree<long, long> CreateRangeAddRangeMax(IEnumerable<long> init) => new LIB_LazySegTree<long, long>(init, long.MinValue, 0, f6, f4, f5);
        static public LIB_LazySegTree<long, long> CreateRangeUpdateRangeSum(IEnumerable<long> init) => new LIB_LazySegTree<long, long>(init, 0, long.MaxValue, f5, f7, f3);
        static public LIB_LazySegTree<long, long> CreateRangeAddRangeSum(IEnumerable<long> init) => new LIB_LazySegTree<long, long>(init, 0, 0, f5, f8, f5);
    }
}