Problem:
Score : 900 points

Problem Statement

We have a tree with N vertices.
The vertices are numbered 0 through N - 1, and the i-th edge (0 ≤ i < N - 1) comnnects Vertex a_i and b_i.
For each pair of vertices u and v (0 ≤ u, v < N), we define the distance d(u, v) as the number of edges in the path u-v.

It is expected that one of the vertices will be invaded by aliens from outer space.
Snuke wants to immediately identify that vertex when the invasion happens.
To do so, he has decided to install an antenna on some vertices.

First, he decides the number of antennas, K (1 ≤ K ≤ N).
Then, he chooses K different vertices, x_0, x_1, ..., x_{K - 1}, on which he installs Antenna 0, 1, ..., K - 1, respectively.
If Vertex v is invaded by aliens, Antenna k (0 ≤ k < K) will output the distance d(x_k, v).
Based on these K outputs, Snuke will identify the vertex that is invaded.
Thus, in order to identify the invaded vertex no matter which one is invaded, the following condition must hold:

For each vertex u (0 ≤ u < N), consider the vector (d(x_0, u), ..., d(x_{K - 1}, u)). These N vectors are distinct.

Find the minumum value of K, the number of antennas, when the condition is satisfied.

Constraints

2 ≤ N ≤ 10^5

0 ≤ a_i, b_i < N

The given graph is a tree.

Input

Input is given from Standard Input in the following format:

N
a_0 b_0
a_1 b_1
:
a_{N - 2} b_{N - 2}

Output

Print the minumum value of K, the number of antennas, when the condition is satisfied.

Sample Input 1

5
0 1
0 2
0 3
3 4

Sample Output 1

2

For example, install an antenna on Vertex 1 and 3.
Then, the following five vectors are distinct:

(d(1, 0), d(3, 0)) = (1, 1)

(d(1, 1), d(3, 1)) = (0, 2)

(d(1, 2), d(3, 2)) = (2, 2)

(d(1, 3), d(3, 3)) = (2, 0)

(d(1, 4), d(3, 4)) = (3, 1)

Sample Input 2

2
0 1

Sample Output 2

1

For example, install an antenna on Vertex 0.

Sample Input 3

10
2 8
6 0
4 1
7 6
2 3
8 6
6 9
2 4
5 8

Sample Output 3

3

For example, install an antenna on Vertex 0, 4, 9.

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 System.Text;
using System.Runtime.CompilerServices;
using System.Runtime.InteropServices;
using Kujikatsu102.Algorithms;
using Kujikatsu102.Collections;
using Kujikatsu102.Numerics;
using Kujikatsu102.Questions;
using Kujikatsu102.Graphs;

namespace Kujikatsu102.Questions
{
    /// <summary>
    /// https://atcoder.jp/contests/apc001/tasks/apc001_e
    /// </summary>
    public class QuestionG : AtCoderQuestionBase
    {
        public override void Solve(IOManager io)
        {
            var n = io.ReadInt();
            var graph = new BasicGraph(n);

            var degrees = new int[n];

            for (int i = 0; i < n - 1; i++)
            {
                var a = io.ReadInt();
                var b = io.ReadInt();
                graph.AddEdge(new BasicEdge(a, b));
                graph.AddEdge(new BasicEdge(b, a));
                degrees[a]++;
                degrees[b]++;
            }

            for (int i = 0; i < degrees.Length; i++)
            {
                if (degrees[i] > 2)
                {
                    var result = Dfs(0, -1);
                    if (graph[0].Count() == 1)
                    {
                        result++;
                    }

                    io.WriteLine(result);
                    return;
                }
            }

            io.WriteLine(1);

            int Dfs(int current, int parent)
            {
                var children = 0;
                var antennas = 0;
                var antennaBranches = 0;

                foreach (var edge in graph[current])
                {
                    var next = edge.To.Index;
                    if (next == parent)
                    {
                        continue;
                    }

                    children++;
                    var an = Dfs(next, current);
                    antennas += an;

                    if (an > 0)
                    {
                        antennaBranches++;
                    }
                }

                if (children > antennaBranches + 1)
                {
                    antennas += children - (antennaBranches + 1);
                }

                return antennas;
            }
        }
    }
}

namespace Kujikatsu102
{
    class Program
    {
        static void Main(string[] args)
        {
            IAtCoderQuestion question = new QuestionG();
            using var io = new IOManager(Console.OpenStandardInput(), Console.OpenStandardOutput());
            question.Solve(io);
        }
    }
}

#region Base Class

namespace Kujikatsu102.Questions
{

    public interface IAtCoderQuestion
    {
        string Solve(string input);
        void Solve(IOManager io);
    }

    public abstract class AtCoderQuestionBase : IAtCoderQuestion
    {
        public string Solve(string input)
        {
            var inputStream = new MemoryStream(Encoding.UTF8.GetBytes(input));
            var outputStream = new MemoryStream();
            using var manager = new IOManager(inputStream, outputStream);

            Solve(manager);
            manager.Flush();

            outputStream.Seek(0, SeekOrigin.Begin);
            var reader = new StreamReader(outputStream);
            return reader.ReadToEnd();
        }

        public abstract void Solve(IOManager io);
    }

    public class IOManager : IDisposable
    {
        private readonly StreamReader _reader;
        private readonly StreamWriter _writer;
        private bool _disposedValue;
        private Queue<ReadOnlyMemory<char>> _stringQueue;

        const char ValidFirstChar = '!';
        const char ValidLastChar = '~';

        public IOManager(Stream input, Stream output)
        {
            _reader = new StreamReader(input);
            _writer = new StreamWriter(output) { AutoFlush = false };
            _stringQueue = new Queue<ReadOnlyMemory<char>>();
        }

        public ReadOnlySpan<char> ReadCharSpan()
        {
            while (_stringQueue.Count == 0)
            {
                var line = _reader.ReadLine().AsMemory().Trim();
                var s = line.Span;

                if (s.Length > 0)
                {
                    var begin = 0;
                    for (int i = 0; i < s.Length; i++)
                    {
                        if (begin < 0 && IsValidChar(s[i]))
                        {
                            begin = i;
                        }
                        else if (!IsValidChar(s[i]))
                        {
                            _stringQueue.Enqueue(line[begin..i]);
                            begin = -1;
                        }
                    }
                    _stringQueue.Enqueue(line[begin..line.Length]);
                }
            }

            return _stringQueue.Dequeue().Span;
        }

        public string ReadString() => ReadCharSpan().ToString();

        public int ReadInt() => (int)ReadLong();

        public long ReadLong()
        {
            long result = 0;
            bool isPositive = true;

            int i = 0;
            var s = ReadCharSpan();
            if (s[i] == '-')
            {
                isPositive = false;
                i++;
            }

            while (i < s.Length)
            {
                result = result * 10 + (s[i++] - '0');
            }

            return isPositive ? result : -result;
        }

        public double ReadDouble() => double.Parse(ReadCharSpan());
        public decimal ReadDecimal() => decimal.Parse(ReadCharSpan());

        public int[] ReadIntArray(int n)
        {
            var a = new int[n];
            for (int i = 0; i < a.Length; i++)
            {
                a[i] = ReadInt();
            }
            return a;
        }

        public long[] ReadLongArray(int n)
        {
            var a = new long[n];
            for (int i = 0; i < a.Length; i++)
            {
                a[i] = ReadLong();
            }
            return a;
        }

        public double[] ReadDoubleArray(int n)
        {
            var a = new double[n];
            for (int i = 0; i < a.Length; i++)
            {
                a[i] = ReadDouble();
            }
            return a;
        }

        public decimal[] ReadDecimalArray(int n)
        {
            var a = new decimal[n];
            for (int i = 0; i < a.Length; i++)
            {
                a[i] = ReadDecimal();
            }
            return a;
        }

        public void WriteLine<T>(T value) => _writer.WriteLine(value.ToString());

        public void WriteLine<T>(IEnumerable<T> values, char separator)
        {
            var e = values.GetEnumerator();
            if (e.MoveNext())
            {
                _writer.Write(e.Current.ToString());

                while (e.MoveNext())
                {
                    _writer.Write(separator);
                    _writer.Write(e.Current.ToString());
                }
            }

            _writer.WriteLine();
        }

        public void Flush() => _writer.Flush();

        private static bool IsValidChar(char c) => ValidFirstChar <= c && c <= ValidLastChar;

        protected virtual void Dispose(bool disposing)
        {
            if (!_disposedValue)
            {
                if (disposing)
                {
                    _reader.Dispose();
                    _writer.Flush();
                    _writer.Dispose();
                }

                _disposedValue = true;
            }
        }

        public void Dispose()
        {
            Dispose(disposing: true);
            GC.SuppressFinalize(this);
        }
    }
}

#endregion

#region Algorithm

namespace Kujikatsu102.Numerics
{
    public static class NumericalAlgorithms
    {
        public static long Gcd(long a, long b)
        {
            if (a < b)
            {
                (a, b) = (b, a);
            }

            if (b > 0)
            {
                return Gcd(b, a % b);
            }
            else if (b == 0)
            {
                return a;
            }
            else
            {
                throw new ArgumentOutOfRangeException($"{nameof(a)}, {nameof(b)}は0以上の整数である必要があります。");
            }
        }

        public static long Lcm(long a, long b)
        {
            if (a < 0 || b < 0)
            {
                throw new ArgumentOutOfRangeException($"{nameof(a)}, {nameof(b)}は0以上の整数である必要があります。");
            }

            return a / Gcd(a, b) * b;
        }

        public static long Factorial(int n)
        {
            if (n < 0)
            {
                throw new ArgumentOutOfRangeException(nameof(n), $"{n}は0以上の整数でなければなりません。");
            }

            long result = 1;
            for (int i = 2; i <= n; i++)
            {
                result *= i;
            }
            return result;
        }

        public static long Permutation(int n, int r)
        {
            CheckNR(n, r);
            long result = 1;
            for (int i = 0; i < r; i++)
            {
                result *= n - i;
            }
            return result;
        }

        public static long Combination(int n, int r)
        {
            CheckNR(n, r);
            r = Math.Min(r, n - r);

            // See https://stackoverflow.com/questions/1838368/calculating-the-amount-of-combinations
            long result = 1;
            for (int i = 1; i <= r; i++)
            {
                result *= n--;
                result /= i;
            }
            return result;
        }

        public static long CombinationWithRepetition(int n, int r) => Combination(n + r - 1, r);

        public static IEnumerable<(int prime, int count)> PrimeFactorization(int n)
        {
            if (n <= 1)
            {
                throw new ArgumentOutOfRangeException(nameof(n), $"{n}は2以上の整数でなければなりません。");
            }

            var dictionary = new Dictionary<int, int>();
            for (int i = 2; i * i <= n; i++)
            {
                while (n % i == 0)
                {
                    if (dictionary.ContainsKey(i))
                    {
                        dictionary[i]++;
                    }
                    else
                    {
                        dictionary[i] = 1;
                    }

                    n /= i;
                }
            }

            if (n > 1)
            {
                dictionary[n] = 1;
            }

            return dictionary.Select(p => (p.Key, p.Value));
        }

        private static void CheckNR(int n, int r)
        {
            if (n <= 0)
            {
                throw new ArgumentOutOfRangeException(nameof(n), $"{nameof(n)}は正の整数でなければなりません。");
            }
            if (r < 0)
            {
                throw new ArgumentOutOfRangeException(nameof(r), $"{nameof(r)}は0以上の整数でなければなりません。");
            }
            if (n < r)
            {
                throw new ArgumentOutOfRangeException($"{nameof(n)},{nameof(r)}", $"{nameof(r)}は{nameof(n)}以下でなければなりません。");
            }
        }
    }

    public readonly struct Modular : IEquatable<Modular>, IComparable<Modular>
    {
        private const int DefaultMod = 1000000007;
        public int Value { get; }
        public static int Mod { get; set; } = DefaultMod;

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public Modular(long value)
        {
            if (unchecked((ulong)value) < unchecked((ulong)Mod))
            {
                Value = (int)value;
            }
            else
            {
                Value = (int)(value % Mod);
                if (Value < 0)
                {
                    Value += Mod;
                }
            }
        }

        private Modular(int value) => Value = value;
        public static Modular Zero => new Modular(0);
        public static Modular One => new Modular(1);

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static Modular operator +(Modular a, Modular b)
        {
            var result = a.Value + b.Value;
            if (result >= Mod)
            {
                result -= Mod;    // 剰余演算を避ける
            }
            return new Modular(result);
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static Modular operator -(Modular a, Modular b)
        {
            var result = a.Value - b.Value;
            if (result < 0)
            {
                result += Mod;    // 剰余演算を避ける
            }
            return new Modular(result);
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static Modular operator *(Modular a, Modular b) => new Modular((long)a.Value * b.Value);

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static Modular operator /(Modular a, Modular b) => a * Pow(b.Value, Mod - 2);

        // 需要は不明だけど一応
        public static bool operator ==(Modular left, Modular right) => left.Equals(right);
        public static bool operator !=(Modular left, Modular right) => !(left == right);
        public static bool operator <(Modular left, Modular right) => left.CompareTo(right) < 0;
        public static bool operator <=(Modular left, Modular right) => left.CompareTo(right) <= 0;
        public static bool operator >(Modular left, Modular right) => left.CompareTo(right) > 0;
        public static bool operator >=(Modular left, Modular right) => left.CompareTo(right) >= 0;

        public static implicit operator Modular(long a) => new Modular(a);
        public static explicit operator int(Modular a) => a.Value;
        public static explicit operator long(Modular a) => a.Value;

        public static Modular Pow(int a, int n)
        {
            if (n == 0)
            {
                return Modular.One;
            }
            else if (n == 1)
            {
                return a;
            }
            else if (n > 0)
            {
                var p = Pow(a, n >> 1);             // m / 2
                return p * p * Pow(a, n & 0x01);    // m % 2
            }
            else
            {
                throw new ArgumentOutOfRangeException(nameof(n), $"べき指数{nameof(n)}は0以上の整数でなければなりません。");
            }
        }

        private static List<int> _factorialCache;
        private static List<int> FactorialCache => _factorialCache ??= new List<int>() { 1 };
        private static int[] FactorialInverseCache { get; set; }
        const int defaultMaxFactorial = 1000000;

        public static Modular Factorial(int n)
        {
            if (n < 0)
            {
                throw new ArgumentOutOfRangeException(nameof(n), $"{nameof(n)}は0以上の整数でなければなりません。");
            }

            for (int i = FactorialCache.Count; i <= n; i++)  // Countが1（0!までキャッシュ済み）のとき1!～n!まで計算
            {
                FactorialCache.Add((int)((long)FactorialCache[i - 1] * i % Mod));
            }
            return new Modular(FactorialCache[n]);
        }

        public static Modular Permutation(int n, int r)
        {
            CheckNR(n, r);
            return Factorial(n) / Factorial(n - r);
        }

        public static Modular Combination(int n, int r)
        {
            CheckNR(n, r);
            r = Math.Min(r, n - r);
            try
            {
                return new Modular(FactorialCache[n]) * new Modular(FactorialInverseCache[r]) * new Modular(FactorialInverseCache[n - r]);
            }
            catch (Exception ex) when (ex is NullReferenceException || ex is ArgumentOutOfRangeException)
            {
                throw new InvalidOperationException($"{nameof(Combination)}を呼び出す前に{nameof(InitializeCombinationTable)}により前計算を行う必要があります。", ex);
            }
        }

        public static void InitializeCombinationTable(int max = defaultMaxFactorial)
        {
            Factorial(max);
            FactorialInverseCache = new int[max + 1];

            var fInv = (Modular.One / Factorial(max)).Value;
            FactorialInverseCache[max] = fInv;
            for (int i = max - 1; i >= 0; i--)
            {
                fInv = (int)((long)fInv * (i + 1) % Mod);
                FactorialInverseCache[i] = fInv;
            }
        }

        public static Modular CombinationWithRepetition(int n, int r) => Combination(n + r - 1, r);

        private static void CheckNR(int n, int r)
        {
            if (n < 0)
            {
                throw new ArgumentOutOfRangeException(nameof(n), $"{nameof(n)}は0以上の整数でなければなりません。");
            }
            if (r < 0)
            {
                throw new ArgumentOutOfRangeException(nameof(r), $"{nameof(r)}は0以上の整数でなければなりません。");
            }
            if (n < r)
            {
                throw new ArgumentOutOfRangeException($"{nameof(n)},{nameof(r)}", $"{nameof(r)}は{nameof(n)}以下でなければなりません。");
            }
        }

        public override string ToString() => Value.ToString();
        public override bool Equals(object obj) => obj is Modular m ? Equals(m) : false;
        public bool Equals([System.Diagnostics.CodeAnalysis.AllowNull] Modular other) => Value == other.Value;
        public int CompareTo([System.Diagnostics.CodeAnalysis.AllowNull] Modular other) => Value.CompareTo(other.Value);
        public override int GetHashCode() => Value.GetHashCode();
    }

    public class ModMatrix
    {
        readonly Modular[] _values;
        public int Height { get; }
        public int Width { get; }

        public Span<Modular> this[int row]
        {
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            get => _values.AsSpan(row * Width, Width);
        }

        public Modular this[int row, int column]
        {
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            get
            {
                if (unchecked((uint)row) >= Height)
                    ThrowsArgumentOutOfRangeException(nameof(row));
                else if (unchecked((uint)column) >= Width)
                    ThrowsArgumentOutOfRangeException(nameof(column));
                return _values[row * Width + column];
            }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            set
            {
                if (unchecked((uint)row) >= Height)
                    ThrowsArgumentOutOfRangeException(nameof(row));
                else if (unchecked((uint)column) >= Width)
                    ThrowsArgumentOutOfRangeException(nameof(column));
                _values[row * Width + column] = value;
            }
        }

        public ModMatrix(int n) : this(n, n) { }

        public ModMatrix(int height, int width)
        {
            if (height <= 0)
                throw new ArgumentOutOfRangeException(nameof(height));
            if (width <= 0)
                throw new ArgumentOutOfRangeException(nameof(width));
            Height = height;
            Width = width;
            _values = new Modular[height * width];
        }

        public ModMatrix(Modular[][] values) : this(values.Length, values[0].Length)
        {
            for (int row = 0; row < Height; row++)
            {
                if (Width != values[row].Length)
                    throw new ArgumentException($"{nameof(values)}の列数は揃っている必要があります。");
                var span = _values.AsSpan(row * Width, Width);
                values[row].AsSpan().CopyTo(span);
            }
        }

        public ModMatrix(Modular[,] values) : this(values.GetLength(0), values.GetLength(1))
        {
            for (int row = 0; row < Height; row++)
            {
                var span = _values.AsSpan(row * Width, Width);
                for (int column = 0; column < span.Length; column++)
                {
                    span[column] = values[row, column];
                }
            }
        }

        public ModMatrix(ModMatrix matrix)
        {
            Height = matrix.Height;
            Width = matrix.Width;
            _values = new Modular[matrix._values.Length];
            matrix._values.AsSpan().CopyTo(_values);
        }

        public static ModMatrix GetIdentity(int dimension)
        {
            var result = new ModMatrix(dimension);
            for (int i = 0; i < dimension; i++)
            {
                result._values[i * result.Width + i] = 1;
            }
            return result;
        }

        public static ModMatrix operator +(ModMatrix a, ModMatrix b)
        {
            CheckSameShape(a, b);

            var result = new ModMatrix(a.Height, a.Width);
            for (int i = 0; i < result._values.Length; i++)
            {
                result._values[i] = a._values[i] + b._values[i];
            }
            return result;
        }

        public static ModMatrix operator -(ModMatrix a, ModMatrix b)
        {
            CheckSameShape(a, b);

            var result = new ModMatrix(a.Height, a.Width);
            for (int i = 0; i < result._values.Length; i++)
            {
                result._values[i] = a._values[i] - b._values[i];
            }
            return result;
        }

        public static ModMatrix operator *(ModMatrix a, ModMatrix b)
        {
            if (a.Width != b.Height)
                throw new ArgumentException($"{nameof(a)}の列数と{nameof(b)}の行数は等しくなければなりません。");

            var result = new ModMatrix(a.Height, b.Width);
            for (int i = 0; i < result.Height; i++)
            {
                var aSpan = a._values.AsSpan(i * a.Width, a.Width);
                var resultSpan = result._values.AsSpan(i * result.Width, result.Width);
                for (int k = 0; k < aSpan.Length; k++)
                {
                    var bSpan = b._values.AsSpan(k * b.Width, b.Width);
                    for (int j = 0; j < resultSpan.Length; j++)
                    {
                        resultSpan[j] += aSpan[k] * bSpan[j];
                    }
                }
            }
            return result;
        }

        public static ModVector operator *(ModMatrix matrix, ModVector vector)
        {
            if (matrix.Width != vector.Length)
                throw new ArgumentException($"{nameof(matrix)}の列数と{nameof(vector)}の行数は等しくなければなりません。");

            var result = new ModVector(vector.Length);
            for (int i = 0; i < result.Length; i++)
            {
                var matrixSpan = matrix[i];
                for (int k = 0; k < matrixSpan.Length; k++)
                {
                    result[i] += matrixSpan[k] * vector[k];
                }
            }
            return result;
        }

        public ModMatrix Pow(long pow)
        {
            if (Height != Width)
                throw new ArgumentException("累乗を行う行列は正方行列である必要があります。");
            if (pow < 0)
                throw new ArgumentException($"{nameof(pow)}は0以上の整数である必要があります。");

            var powMatrix = new ModMatrix(this);
            var result = GetIdentity(Height);
            while (pow > 0)
            {
                if ((pow & 1) > 0)
                {
                    result *= powMatrix;
                }
                powMatrix *= powMatrix;
                pow >>= 1;
            }
            return result;
        }

        private static void CheckSameShape(ModMatrix a, ModMatrix b)
        {
            if (a.Height != b.Height)
                throw new ArgumentException($"{nameof(a)}の行数と{nameof(b)}の行数は等しくなければなりません。");
            else if (a.Width != b.Width)
                throw new ArgumentException($"{nameof(a)}の列数と{nameof(b)}の列数は等しくなければなりません。");
        }

        private void ThrowsArgumentOutOfRangeException(string paramName) => throw new ArgumentOutOfRangeException(paramName);
        public override string ToString() => $"({Height}x{Width})matrix";
    }

    public class ModVector
    {
        readonly Modular[] _values;
        public int Length => _values.Length;

        public ModVector(int length)
        {
            if (length <= 0)
                throw new ArgumentOutOfRangeException(nameof(length));
            _values = new Modular[length];
        }

        public ModVector(ReadOnlySpan<Modular> vector)
        {
            _values = new Modular[vector.Length];
            vector.CopyTo(_values);
        }

        public ModVector(ModVector vector) : this(vector._values) { }

        public Modular this[int index]
        {
            get => _values[index];
            set => _values[index] = value;
        }

        public static ModVector operator +(ModVector a, ModVector b)
        {
            if (a.Length != b.Length)
                throw new ArgumentException($"{nameof(a)}と{nameof(b)}の次元は等しくなければなりません。");

            var result = new ModVector(a.Length);
            for (int i = 0; i < result.Length; i++)
            {
                result[i] = a[i] + b[i];
            }
            return result;
        }

        public static ModVector operator -(ModVector a, ModVector b)
        {
            if (a.Length != b.Length)
                throw new ArgumentException($"{nameof(a)}と{nameof(b)}の次元は等しくなければなりません。");

            var result = new ModVector(a.Length);
            for (int i = 0; i < result.Length; i++)
            {
                result[i] = a[i] - b[i];
            }
            return result;
        }

        public static Modular operator *(ModVector a, ModVector b)
        {
            if (a.Length != b.Length)
                throw new ArgumentException($"{nameof(a)}と{nameof(b)}の次元は等しくなければなりません。");

            var result = Modular.Zero;
            for (int i = 0; i < a.Length; i++)
            {
                result += a[i] * b[i];
            }
            return result;
        }

        public override string ToString() => $"({Length})vector";
    }

    public class Eratosthenes
    {
        /// <summary>
        /// Smallest Prime Factorを保存した配列
        /// </summary>
        readonly int[] _spf;

        public Eratosthenes(int max)
        {
            _spf = Enumerable.Range(0, max + 1).ToArray();
            for (int i = 2; i * i <= max; i++)
            {
                if (_spf[i] == i)
                {
                    for (int mul = i << 1; mul <= max; mul += i)
                    {
                        if (_spf[mul] == mul)
                        {
                            _spf[mul] = i;
                        }
                    }
                }
            }
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public bool IsPrime(int n) => n >= 2 && _spf[n] == n;

        public IEnumerable<PrimeAndCount> PrimeFactorize(int n)
        {
            if (n <= 0 || _spf.Length <= n)
            {
                throw new ArgumentOutOfRangeException(nameof(n), $"{nameof(n)}は[0, {_spf.Length})の間でなければなりません");
            }
            else if (n == 1)
            {
                yield break;
            }
            else
            {
                var last = _spf[n];
                var streak = 0;
                while (n > 1)
                {
                    if (_spf[n] == last)
                    {
                        streak++;
                    }
                    else
                    {
                        yield return new PrimeAndCount(last, streak);
                        last = _spf[n];
                        streak = 1;
                    }

                    n /= last;
                }
                yield return new PrimeAndCount(last, streak);
            }
        }

        public IEnumerable<int> GetDivisiors(int n)
        {
            if (n <= 0 || _spf.Length <= n)
            {
                throw new ArgumentOutOfRangeException(nameof(n), $"{nameof(n)}は[0, {_spf.Length})の間でなければなりません");
            }
            else
            {
                var primes = PrimeFactorize(n).ToArray();
                return GetDivisiors(primes, 0);
            }
        }

        IEnumerable<int> GetDivisiors(PrimeAndCount[] primes, int depth)
        {
            if (depth == primes.Length)
            {
                yield return 1;
            }
            else
            {
                var current = 1;
                var children = GetDivisiors(primes, depth + 1).ToArray();

                foreach (var child in children)
                {
                    yield return child;
                }

                for (int i = 0; i < primes[depth].Count; i++)
                {
                    current *= primes[depth].Prime;
                    foreach (var child in children)
                    {
                        yield return current * child;
                    }
                }
            }
        }
    }

    public static class Divisiors
    {
        public static IEnumerable<long> GetDivisiors(long n)
        {
            var lastHalf = new Stack<long>();
            for (long i = 1; i * i <= n; i++)
            {
                if (n % i == 0)
                {
                    yield return i;
                    if (i * i != n)
                    {
                        lastHalf.Push(n / i);
                    }
                }
            }

            while (lastHalf.Count > 0)
            {
                yield return lastHalf.Pop();
            }
        }
    }

    [StructLayout(LayoutKind.Auto)]
    public readonly struct PrimeAndCount
    {
        public int Prime { get; }
        public int Count { get; }

        public PrimeAndCount(int prime, int count)
        {
            Prime = prime;
            Count = count;
        }

        public void Deconstruct(out int prime, out int count) => (prime, count) = (Prime, Count);
        public override string ToString() => $"{nameof(Prime)}: {Prime}, {nameof(Count)}: {Count}";
    }

    public readonly struct Fraction : IEquatable<Fraction>, IComparable<Fraction>
    {
        /// <summary>分子</summary>
        public long Numerator { get; }
        /// <summary>分母</summary>
        public long Denominator { get; }

        public static Fraction Nan => new Fraction(0, 0);
        public static Fraction PositiveInfinity => new Fraction(1, 0);
        public static Fraction NegativeInfinity => new Fraction(-1, 0);
        public bool IsNan => Numerator == 0 && Denominator == 0;
        public bool IsInfinity => Numerator != 0 && Denominator == 0;
        public bool IsPositiveInfinity => Numerator > 0 && Denominator == 0;
        public bool IsNegativeInfinity => Numerator < 0 && Denominator == 0;

        /// <summary>
        /// <c>Fraction</c>クラスの新しいインスタンスを生成します。
        /// </summary>
        /// <param name="numerator">分子</param>
        /// <param name="denominator">分母</param>
        public Fraction(long numerator, long denominator)
        {
            if (denominator == 0)
            {
                Numerator = Math.Sign(numerator);
                Denominator = 0;
            }
            else if (numerator == 0)
            {
                Numerator = 0;
                Denominator = 1;
            }
            else
            {
                var sign = Math.Sign(numerator) * Math.Sign(denominator);
                numerator = Math.Abs(numerator);
                denominator = Math.Abs(denominator);
                var gcd = NumericalAlgorithms.Gcd(numerator, denominator);
                Numerator = sign * numerator / gcd;
                Denominator = denominator / gcd;
            }
        }

        public static Fraction operator +(in Fraction left, in Fraction right)
        {
            if (left.IsNan || right.IsNan)
            {
                return Nan;
            }
            else if (left.IsInfinity || right.IsInfinity)
            {
                if (!right.IsInfinity)
                {
                    return left;
                }
                else if (!left.IsInfinity)
                {
                    return right;
                }
                else
                {
                    return new Fraction(left.Numerator + right.Numerator, 0);
                }
            }
            else
            {
                var lcm = NumericalAlgorithms.Lcm(left.Denominator, right.Denominator);
                return new Fraction(left.Numerator * (lcm / left.Denominator) + right.Numerator * (lcm / right.Denominator), lcm);
            }
        }
        public static Fraction operator -(in Fraction left, in Fraction right) => left + -right;
        public static Fraction operator *(in Fraction left, in Fraction right) => new Fraction(left.Numerator * right.Numerator, left.Denominator * right.Denominator);
        public static Fraction operator /(in Fraction left, in Fraction right) => new Fraction(left.Numerator * right.Denominator, left.Denominator * right.Numerator);
        public static Fraction operator +(in Fraction right) => right;
        public static Fraction operator -(in Fraction right) => new Fraction(-right.Numerator, right.Denominator);
        public static bool operator ==(in Fraction left, in Fraction right) => left.Equals(right);
        public static bool operator !=(in Fraction left, in Fraction right) => !(left == right);
        public static implicit operator double(in Fraction right)
        {
            if (right.IsNan)
            {
                return double.NaN;
            }
            else if (right.IsPositiveInfinity)
            {
                return double.PositiveInfinity;
            }
            else if (right.IsNegativeInfinity)
            {
                return double.NegativeInfinity;
            }
            else
            {
                return (double)right.Numerator / right.Denominator;
            }
        }

        public override string ToString()
        {
            if (IsNan)
            {
                return "NaN";
            }
            else if (IsPositiveInfinity)
            {
                return "Inf";
            }
            else if (IsNegativeInfinity)
            {
                return "-Inf";
            }
            else
            {
                return $"{Numerator}/{Denominator}";
            }
        }

        public override bool Equals(object obj) => obj is Fraction fraction && Equals(fraction);
        public bool Equals(Fraction other) => Numerator == other.Numerator && Denominator == other.Denominator;
        public override int GetHashCode() => HashCode.Combine(Numerator, Denominator);
        public int CompareTo([System.Diagnostics.CodeAnalysis.AllowNull] Fraction other) => ((double)this).CompareTo(other);
    }

    public interface ISemigroup<TSet> where TSet : ISemigroup<TSet>
    {
        public TSet Merge(TSet other);
    }

    public interface IMonoid<TSet> : ISemigroup<TSet> where TSet : IMonoid<TSet>, new()
    {
        public TSet Identity { get; }
    }

    public interface IMonoidWithAct<TMonoid, TOperator> : IMonoid<TOperator>
        where TMonoid : IMonoid<TMonoid>, new()
        where TOperator : IMonoid<TOperator>, new()
    {
        public TMonoid Act(TMonoid monoid);
    }

    public interface IGroup<TSet> : IMonoid<TSet> where TSet : IGroup<TSet>, new()
    {
        public TSet Invert();
        public static TSet operator ~(IGroup<TSet> a) => a.Invert();
    }
}

namespace Kujikatsu102.Algorithms
{
    public static class ZAlgorithm
    {
        public static int[] SearchAll(string s) => SearchAll(s.AsSpan());

        public static int[] SearchAll<T>(ReadOnlySpan<T> s) where T : IEquatable<T>
        {
            var z = new int[s.Length];
            z[0] = s.Length;
            var offset = 1;
            var length = 0;

            while (offset < s.Length)
            {
                while (offset + length < s.Length && s[length].Equals(s[offset + length]))
                {
                    length++;
                }
                z[offset] = length;

                if (length == 0)
                {
                    offset++;
                    continue;
                }

                int copyLength = 1;
                while (copyLength < length && copyLength + z[copyLength] < length)
                {
                    z[offset + copyLength] = z[copyLength];
                    copyLength++;
                }
                offset += copyLength;
                length -= copyLength;
            }

            return z;
        }
    }

    /// <summary>
    /// MP法（文字列検索アルゴリズム）
    /// </summary>
    public class MorrisPratt<T> where T : IEquatable<T>
    {
        readonly T[] _searchSequence;
        readonly int[] _matchLength;

        public ReadOnlySpan<T> SearchSequence => _searchSequence.AsSpan();

        /// <summary>
        /// 検索データ列の前処理を行います。
        /// </summary>
        /// <param name="searchSequence">検索データ列</param>
        public MorrisPratt(ReadOnlySpan<T> searchSequence)
        {
            _searchSequence = searchSequence.ToArray();
            _matchLength = new int[_searchSequence.Length + 1];
            _matchLength[0] = -1;
            int j = -1;
            for (int i = 0; i < _searchSequence.Length; i++)
            {
                while (j != -1 && !_searchSequence[j].Equals(_searchSequence[i]))
                {
                    j = _matchLength[j];
                }
                j++;
                _matchLength[i + 1] = j;
            }
        }

        /// <summary>
        /// 与えられた対象データ列の部分列のうち、検索データ列にマッチする部分列の開始インデックスを取得します。
        /// </summary>
        /// <param name="targetSequence">検索対象データ列</param>
        /// <returns></returns>
        public List<int> SearchAll(ReadOnlySpan<T> targetSequence)
        {
            var results = new List<int>();
            int j = 0;
            for (int i = 0; i < targetSequence.Length; i++)
            {
                while (j != -1 && !_searchSequence[j].Equals(targetSequence[i]))
                {
                    j = _matchLength[j];
                }
                j++;
                if (j == _searchSequence.Length)
                {
                    results.Add(i - j + 1);
                    j = _matchLength[j];
                }
            }
            return results;
        }
    }

    /// <summary>
    /// 参考: https://qiita.com/keymoon/items/11fac5627672a6d6a9f6
    /// ジェネリクスに対応させるにはGetHashCode()を足していく？実装によっては重そうなのでとりあえずパス。
    /// </summary>
    public class RollingHash
    {
        const ulong Mask30 = (1UL << 30) - 1;
        const ulong Mask31 = (1UL << 31) - 1;
        const ulong Mod = (1UL << 61) - 1;
        const ulong Positivizer = Mod * ((1UL << 3) - 1);   // 引き算する前に足すことでmodが負になることを防ぐやつ
        static readonly uint base1;
        static readonly uint base2;
        static readonly List<ulong> pow1;
        static readonly List<ulong> pow2;

        static RollingHash()
        {
            var random = new Random();
            base1 = (uint)random.Next(129, int.MaxValue >> 2);
            base2 = (uint)random.Next(int.MaxValue >> 2, int.MaxValue); // 32bit目は0にしておく
            pow1 = new List<ulong>() { 1 };
            pow2 = new List<ulong>() { 1 };
        }

        ulong[] hash1;
        ulong[] hash2;
        public string RawString { get; }
        public int Length => RawString.Length;

        public RollingHash(string s)
        {
            RawString = s;
            hash1 = new ulong[s.Length + 1];
            hash2 = new ulong[s.Length + 1];

            for (int i = pow1.Count; i < s.Length + 1; i++)
            {
                pow1.Add(CalculateModular(Multiply(pow1[i - 1], base1)));
                pow2.Add(CalculateModular(Multiply(pow2[i - 1], base2)));
            }

            for (int i = 0; i < s.Length; i++)
            {
                hash1[i + 1] = CalculateModular(Multiply(hash1[i], base1) + s[i]);
                hash2[i + 1] = CalculateModular(Multiply(hash2[i], base2) + s[i]);
            }
        }

        public (ulong, ulong) this[Range range]
        {
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            get
            {
                var (offset, length) = range.GetOffsetAndLength(Length);
                return Slice(offset, length);
            }
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public (ulong, ulong) Slice(int begin, int length)
        {
            var result1 = CalculateModular(hash1[begin + length] + Positivizer - Multiply(hash1[begin], pow1[length]));
            var result2 = CalculateModular(hash2[begin + length] + Positivizer - Multiply(hash2[begin], pow2[length]));
            return (result1, result2);
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        private static ulong Multiply(ulong l, ulong r)
        {
            var lu = l >> 31;
            var ll = l & Mask31;
            var ru = r >> 31;
            var rl = r & Mask31;
            var mid = ll * ru + lu * rl;
            return ((lu * ru) << 1) + ll * rl + ((mid & Mask30) << 31) + (mid >> 30);   // a * 2^61 ≡ a (mod 2^61 - 1)を使う
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        private static ulong Multiply(ulong l, uint r)
        {
            var lu = l >> 31;
            var mid = lu * r;
            return (l & Mask31) * r + ((mid & Mask30) << 31) + (mid >> 30); // rの32bit目は0としている
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        private static ulong CalculateModular(ulong value)
        {
            value = (value & Mod) + (value >> 61);
            if (value >= Mod)
            {
                value -= Mod;
            }
            return value;
        }

        public override string ToString() => RawString;
    }

    public class XorShift
    {
        ulong _x;

        public XorShift() : this((ulong)DateTime.Now.Ticks) { }

        public XorShift(ulong seed)
        {
            _x = seed;
        }

        /// <summary>
        /// [0, (2^64)-1)の乱数を生成します。
        /// </summary>
        /// <returns></returns>
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public ulong Next()
        {
            _x = _x ^ (_x << 13);
            _x = _x ^ (_x >> 7);
            _x = _x ^ (_x << 17);
            return _x;
        }

        /// <summary>
        /// [0, <c>exclusiveMax</c>)の乱数を生成します。
        /// </summary>
        /// <param name="exclusiveMax"></param>
        /// <returns></returns>
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public int Next(int exclusiveMax) => (int)(Next() % (uint)exclusiveMax);

        /// <summary>
        /// [0.0, 1.0)の乱数を生成します。
        /// </summary>
        /// <returns></returns>
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public double NextDouble()
        {
            const ulong max = 1UL << 50;
            const ulong mask = max - 1;
            return (double)(Next() & mask) / max;
        }
    }

    public static class AlgorithmHelpers
    {
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static void UpdateWhenSmall<T>(ref T value, T other) where T : IComparable<T>
        {
            if (other.CompareTo(value) < 0)
            {
                value = other;
            }
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public static void UpdateWhenLarge<T>(ref T value, T other) where T : IComparable<T>
        {
            if (other.CompareTo(value) > 0)
            {
                value = other;
            }
        }
    }

    public class CoordinateShrinker<T> : IEnumerable<(int shrinkedIndex, T rawIndex)> where T : IComparable<T>, IEquatable<T>
    {
        Dictionary<T, int> _shrinkMapper;
        T[] _expandMapper;
        public int Count => _expandMapper.Length;

        public CoordinateShrinker(IEnumerable<T> data)
        {
            _expandMapper = data.Distinct().ToArray();
            Array.Sort(_expandMapper);

            _shrinkMapper = new Dictionary<T, int>();
            for (int i = 0; i < _expandMapper.Length; i++)
            {
                _shrinkMapper.Add(_expandMapper[i], i);
            }
        }

        public int Shrink(T rawCoordinate) => _shrinkMapper[rawCoordinate];
        public T Expand(int shrinkedCoordinate) => _expandMapper[shrinkedCoordinate];

        public IEnumerator<(int shrinkedIndex, T rawIndex)> GetEnumerator()
        {
            for (int i = 0; i < _expandMapper.Length; i++)
            {
                yield return (i, _expandMapper[i]);
            }
        }

        System.Collections.IEnumerator System.Collections.IEnumerable.GetEnumerator() => GetEnumerator();
    }
}

#endregion

#region Collections

namespace Kujikatsu102.Collections
{
    // See https://kumikomiya.com/competitive-programming-with-c-sharp/
    public class UnionFindTree
    {
        private UnionFindNode[] _nodes;
        public int Count => _nodes.Length;
        public int Groups { get; private set; }

        public UnionFindTree(int count)
        {
            _nodes = Enumerable.Range(0, count).Select(i => new UnionFindNode(i)).ToArray();
            Groups = _nodes.Length;
        }

        public void Unite(int index1, int index2)
        {
            var succeed = _nodes[index1].Unite(_nodes[index2]);
            if (succeed)
            {
                Groups--;
            }
        }

        public bool IsInSameGroup(int index1, int index2) => _nodes[index1].IsInSameGroup(_nodes[index2]);
        public int GetGroupSizeOf(int index) => _nodes[index].GetGroupSize();

        private class UnionFindNode
        {
            private int _height;        // rootのときのみ有効
            private int _groupSize;     // 同上
            private UnionFindNode _parent;
            public int ID { get; }

            public UnionFindNode(int id)
            {
                _height = 0;
                _groupSize = 1;
                _parent = this;
                ID = id;
            }

            public UnionFindNode FindRoot()
            {
                if (_parent != this) // not ref equals
                {
                    var root = _parent.FindRoot();
                    _parent = root;
                }

                return _parent;
            }

            public int GetGroupSize() => FindRoot()._groupSize;

            public bool Unite(UnionFindNode other)
            {
                var thisRoot = this.FindRoot();
                var otherRoot = other.FindRoot();

                if (thisRoot == otherRoot)
                {
                    return false;
                }

                if (thisRoot._height < otherRoot._height)
                {
                    thisRoot._parent = otherRoot;
                    otherRoot._groupSize += thisRoot._groupSize;
                    otherRoot._height = Math.Max(thisRoot._height + 1, otherRoot._height);
                    return true;
                }
                else
                {
                    otherRoot._parent = thisRoot;
                    thisRoot._groupSize += otherRoot._groupSize;
                    thisRoot._height = Math.Max(otherRoot._height + 1, thisRoot._height);
                    return true;
                }
            }

            public bool IsInSameGroup(UnionFindNode other) => this.FindRoot() == other.FindRoot();

            public override string ToString() => $"{ID} root:{FindRoot().ID}";
        }
    }

    public class Deque<T> : IReadOnlyCollection<T>
    {
        public int Count { get; private set; }
        private T[] _data;
        private int _first;
        private int _mask;

        public Deque() : this(4) { }

        public Deque(int minCapacity)
        {
            if (minCapacity <= 0)
            {
                throw new ArgumentOutOfRangeException(nameof(minCapacity), $"{nameof(minCapacity)}は0より大きい値でなければなりません。");
            }
            var capacity = GetPow2Over(minCapacity);
            _data = new T[capacity];
            _first = 0;
            _mask = capacity - 1;
        }

        public Deque(IEnumerable<T> collection)
        {
            var dataArray = collection.ToArray();
            var capacity = GetPow2Over(dataArray.Length);
            _data = new T[capacity];
            _first = 0;
            _mask = capacity - 1;

            for (int i = 0; i < dataArray.Length; i++)
            {
                _data[i] = dataArray[i];
                Count++;
            }
        }

        public T this[Index index]
        {
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            get
            {
                var offset = index.GetOffset(Count);
                if (unchecked((uint)offset) >= Count)
                {
                    ThrowArgumentOutOfRangeException(nameof(index), $"{nameof(index)}がコレクションの範囲外です。");
                }
                return _data[(_first + offset) & _mask];
            }
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void EnqueueFirst(T item)
        {
            if (_data.Length == Count)
            {
                Resize();
            }

            _first = (_first - 1) & _mask;
            _data[_first] = item;
            Count++;
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void EnqueueLast(T item)
        {
            if (_data.Length == Count)
            {
                Resize();
            }

            _data[(_first + Count++) & _mask] = item;
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public T DequeueFirst()
        {
            if (Count == 0)
            {
                ThrowInvalidOperationException("Queueが空です。");
            }

            var value = _data[_first];
            _data[_first++] = default;
            _first &= _mask;
            Count--;
            return value;
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public T DequeueLast()
        {
            if (Count == 0)
            {
                ThrowInvalidOperationException("Queueが空です。");
            }

            var index = (_first + --Count) & _mask;
            var value = _data[index];
            _data[index] = default;
            return value;
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public T PeekFirst()
        {
            if (Count == 0)
            {
                ThrowInvalidOperationException("Queueが空です。");
            }

            return _data[_first];
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public T PeekLast()
        {
            if (Count == 0)
            {
                ThrowInvalidOperationException("Queueが空です。");
            }

            return _data[(_first + Count - 1) & _mask];
        }

        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        private void Resize()
        {
            var newArray = new T[_data.Length << 1];
            var span = _data.AsSpan();
            var firstHalf = span[_first..];
            var lastHalf = span[.._first];
            firstHalf.CopyTo(newArray);
            lastHalf.CopyTo(newArray.AsSpan(firstHalf.Length));
            _data = newArray;
            _first = 0;
            _mask = _data.Length - 1;
        }

        private void ThrowArgumentOutOfRangeException(string paramName, string message) => throw new ArgumentOutOfRangeException(paramName, message);
        private void ThrowInvalidOperationException(string message) => throw new InvalidOperationException(message);

        private int GetPow2Over(int n)
        {
            n--;
            var result = 1;
            while (n != 0)
            {
                n >>= 1;
                result <<= 1;
            }
            return result;
        }

        public IEnumerator<T> GetEnumerator()
        {
            for (int i = 0; i < Count; i++)
            {
                var offset = (_first + i) & _mask;
                yield return _data[offset];
            }
        }

        System.Collections.IEnumerator System.Collections.IEnumerable.GetEnumerator() => GetEnumerator();
    }

    public class PriorityQueue<T> : IEnumerable<T> where T : IComparable<T>
    {
        private List<T> _heap = new List<T>();
        private readonly int _reverseFactor;
        public int Count => _heap.Count;
        public bool IsDescending => _reverseFactor == 1;

        public PriorityQueue(bool descending) : this(descending, null) { }

        public PriorityQueue(bool descending, IEnumerable<T> collection)
        {
            _reverseFactor = descending ? 1 : -1;
            _heap = new List<T>();

            if (collection != null)
            {
                foreach (var item in collection)
                {
                    Enqueue(item);
                }
            }
        }

        public void Enqueue(T item)
        {
            _heap.Add(item);
            UpHeap();
        }

        public T Dequeue()
        {
            var item = _heap[0];
            DownHeap();
            return item;
        }

        public T Peek() => _heap[0];

        private void UpHeap()
        {
            var child = Count - 1;
            while (child > 0)
            {
                int parent = (child - 1) / 2;

                if (Compare(_heap[child], _heap[parent]) > 0)
                {
                    SwapAt(child, parent);
                    child = parent;
                }
                else
                {
                    break;
                }
            }
        }

        private void DownHeap()
        {
            _heap[0] = _heap[Count - 1];
            _heap.RemoveAt(Count - 1);

            var parent = 0;
            while (true)
            {
                var leftChild = 2 * parent + 1;

                if (leftChild > Count - 1)
                {
                    break;
                }

                var target = (leftChild < Count - 1) && (Compare(_heap[leftChild], _heap[leftChild + 1]) < 0) ? leftChild + 1 : leftChild;

                if (Compare(_heap[parent], _heap[target]) < 0)
                {
                    SwapAt(parent, target);
                }
                else
                {
                    break;
                }

                parent = target;
            }
        }

        private int Compare(T a, T b) => _reverseFactor * a.CompareTo(b);

        private void SwapAt(int n, int m) => (_heap[n], _heap[m]) = (_heap[m], _heap[n]);

        public IEnumerator<T> GetEnumerator()
        {
            var copy = new List<T>(_heap);
            try
            {
                while (Count > 0)
                {
                    yield return Dequeue();
                }
            }
            finally
            {
                _heap = copy;
            }
        }

        System.Collections.IEnumerator System.Collections.IEnumerable.GetEnumerator() => GetEnumerator();
    }

    public class SegmentTree<TMonoid> : IEnumerable<TMonoid> where TMonoid : IMonoid<TMonoid>, new()
    {
        private readonly TMonoid[] _data;
        private readonly TMonoid _identityElement;

        private readonly int _leafOffset;   // n - 1
        private readonly int _leafLength;   // n (= 2^k)

        public int Length { get; }          // 実データ長
        public ReadOnlySpan<TMonoid> Data => _data.AsSpan(_leafOffset, Length);

        public SegmentTree(ICollection<TMonoid> data)
        {
            Length = data.Count;
            _leafLength = GetMinimumPow2(data.Count);
            _leafOffset = _leafLength - 1;
            _data = new TMonoid[_leafOffset + _leafLength];
            _identityElement = new TMonoid().Identity;

            data.CopyTo(_data, _leafOffset);
            BuildTree();
        }

        public TMonoid this[int index]
        {
            get => Data[index];
            set
            {
                if (index < 0 || index >= Length)
                {
                    throw new IndexOutOfRangeException($"{nameof(index)}がデータの範囲外です。");
                }
                index += _leafOffset;
                _data[index] = value;
                while (index > 0)
                {
                    // 一つ上の親の更新
                    index = (index - 1) / 2;
                    _data[index] = _data[index * 2 + 1].Merge(_data[index * 2 + 2]);
                }
            }
        }

        public TMonoid Query(Range range)
        {
            var (offset, length) = range.GetOffsetAndLength(Length);
            if (length <= 0)
            {
                throw new ArgumentOutOfRangeException(nameof(range), $"{nameof(range)}の長さは0より大きくなければなりません。");
            }
            return Query(offset, offset + length);
        }

        public TMonoid Query(int begin, int end)
        {
            if (begin < 0)
            {
                throw new ArgumentOutOfRangeException(nameof(begin), $"{nameof(begin)}は0以上の数でなければなりません。");
            }
            if (end > Length)
            {
                throw new ArgumentOutOfRangeException(nameof(end), $"{nameof(end)}は{nameof(Length)}以下でなければなりません。");
            }
            if (begin >= end)
            {
                throw new ArgumentException($"{nameof(begin)},{nameof(end)}", $"{nameof(end)}は{nameof(begin)}より大きい数でなければなりません。");
            }
            return Query(begin, end, 0, 0, _leafLength);
        }

        private TMonoid Query(int begin, int end, int index, int left, int right)
        {
            if (right <= begin || end <= left)      // 範囲外
            {
                return _identityElement;
            }
            else if (begin <= left && right <= end) // 全部含まれる
            {
                return _data[index];
            }
            else    // 一部だけ含まれる
            {
                var leftValue = Query(begin, end, index * 2 + 1, left, (left + right) / 2);     // 左の子
                var rightValue = Query(begin, end, index * 2 + 2, (left + right) / 2, right);   // 右の子
                return leftValue.Merge(rightValue);
            }
        }

        private void BuildTree()
        {
            foreach (ref var unusedLeaf in _data.AsSpan()[(_leafOffset + Length)..])
            {
                unusedLeaf = _identityElement;  // 単位元埋め
            }

            for (int i = _leafLength - 2; i >= 0; i--)  // 葉の親から順番に一つずつ上がっていく
            {
                _data[i] = _data[2 * i + 1].Merge(_data[2 * i + 2]); // f(left, right)
            }
        }

        private int GetMinimumPow2(int n)
        {
            var p = 1;
            while (p < n)
            {
                p *= 2;
            }
            return p;
        }

        public IEnumerator<TMonoid> GetEnumerator()
        {
            var upperIndex = _leafOffset + Length;
            for (int i = _leafOffset; i < upperIndex; i++)
            {
                yield return _data[i];
            }
        }

        System.Collections.IEnumerator System.Collections.IEnumerable.GetEnumerator() => GetEnumerator();
    }

    public class LazySegmentTree<TMonoid, TOperator>
        where TMonoid : IMonoid<TMonoid>, new()
        where TOperator : IMonoidWithAct<TMonoid, TOperator>, IEquatable<TOperator>, new()
    {
        private readonly TMonoid[] _data;
        private readonly TOperator[] _lazy;
        private readonly TMonoid _monoidIdenty;
        private readonly TOperator _operatorIdentity;

        private readonly int _leafOffset;  // n - 1
        private readonly int _leafLength;  // n (= 2^k)

        public int Length { get; }

        public LazySegmentTree(ICollection<TMonoid> data)
        {
            Length = data.Count;
            _leafLength = GetMinimumPow2(data.Count);
            _leafOffset = _leafLength - 1;
            _data = new TMonoid[_leafOffset + _leafLength];
            _monoidIdenty = new TMonoid().Identity;
            _operatorIdentity = new TOperator().Identity;

            data.CopyTo(_data, _leafOffset);
            BuildTree();
            _lazy = Enumerable.Repeat(_operatorIdentity, _data.Length).ToArray();
        }

        private void LazyEvaluate(int index)
        {
            if (_lazy[index].Equals(_operatorIdentity))
            {
                return;
            }
            else if (index < _leafOffset) // 葉でない場合は子に伝播
            {
                var left = (index << 1) + 1;
                var right = left + 1;
                _lazy[left] = _lazy[index].Merge(_lazy[left]);
                _lazy[right] = _lazy[index].Merge(_lazy[right]);
            }

            // 自身を更新
            _data[index] = _lazy[index].Act(_data[index]);
            _lazy[index] = _operatorIdentity;
        }

        public void Update(int begin, int end, TOperator op)
        {
            if (begin < 0)
            {
                throw new ArgumentOutOfRangeException(nameof(begin));
            }
            if (end > Length)
            {
                throw new ArgumentOutOfRangeException(nameof(end));
            }
            if (begin >= end)
            {
                throw new ArgumentException($"{nameof(end)} must be grater than {nameof(begin)}");
            }
            Update(begin, end, op, 0, 0, _leafLength);
        }

        private void Update(int begin, int end, TOperator op, int index, int left, int right)
        {
            LazyEvaluate(index);
            if (begin <= left && right <= end) // 全部含まれる
            {
                _lazy[index] = _lazy[index].Merge(op);
                LazyEvaluate(index);
            }
            else if (begin < right && left < end) // 一部だけ含まれる
            {
                var l = (index << 1) + 1;
                var r = l + 1;
                Update(begin, end, op, l, left, (left + right) / 2);
                Update(begin, end, op, r, (left + right) / 2, right);
                _data[index] = _data[l].Merge(_data[r]);
            }
        }

        public TMonoid Query(int begin, int end)
        {
            if (begin < 0)
            {
                throw new ArgumentOutOfRangeException(nameof(begin));
            }
            if (end > Length)
            {
                throw new ArgumentOutOfRangeException(nameof(end));
            }
            if (begin >= end)
            {
                throw new ArgumentException($"{nameof(end)} must be grater than {nameof(begin)}");
            }
            return Query(begin, end, 0, 0, _leafLength);
        }

        private TMonoid Query(int begin, int end, int index, int left, int right)
        {
            LazyEvaluate(index);
            if (right <= begin || end <= left)      // 範囲外
            {
                return _monoidIdenty;
            }
            else if (begin <= left && right <= end) // 全部含まれる
            {
                return _data[index];
            }
            else    // 一部だけ含まれる
            {
                var l = (index << 1) + 1;
                var r = l + 1;
                var leftValue = Query(begin, end, l, left, (left + right) / 2);     // 左の子
                var rightValue = Query(begin, end, r, (left + right) / 2, right);   // 右の子
                return leftValue.Merge(rightValue);
            }
        }

        private void BuildTree()
        {
            foreach (ref var unusedLeaf in _data.AsSpan()[(_leafOffset + Length)..])
            {
                unusedLeaf = _monoidIdenty;  // 単位元埋め
            }

            for (int i = _leafLength - 2; i >= 0; i--)  // 葉の親から順番に一つずつ上がっていく
            {
                var left = (i << 1) + 1;
                var right = left + 1;
                _data[i] = _data[left].Merge(_data[right]); // f(left, right)
            }
        }

        private int GetMinimumPow2(int n)
        {
            var p = 1;
            while (p < n)
            {
                p <<= 1;
            }
            return p;
        }
    }

    public class BinaryIndexedTree
    {
        long[] _data;
        public int Length { get; }

        public BinaryIndexedTree(int length)
        {
            _data = new long[length + 1];   // 内部的には1-indexedにする
            Length = length;
        }

        public BinaryIndexedTree(IEnumerable<long> data, int length) : this(length)
        {
            var count = 0;
            foreach (var n in data)
            {
                AddAt(count++, n);
            }
        }

        public BinaryIndexedTree(ICollection<long> collection) : this(collection, collection.Count) { }

        public long this[Index index]
        {
            get => Sum(index..(index.GetOffset(Length) + 1));
            set
            {
                if (value < 0)
                {
                    throw new ArgumentOutOfRangeException(nameof(value), $"{nameof(value)}は0以上の値である必要があります。");
                }
                AddAt(index, value - this[index]);
            }
        }

        /// <summary>
        /// BITの<c>index</c>番目の要素に<c>n</c>を加算します。
        /// </summary>
        /// <param name="index">加算するインデックス（0-indexed）</param>
        /// <param name="value">加算する数</param>
        public void AddAt(Index index, long value)
        {
            var i = index.GetOffset(Length);
            unchecked
            {
                if ((uint)i >= (uint)Length)
                {
                    throw new ArgumentOutOfRangeException(nameof(index));
                }
            }

            i++;  // 1-indexedにする

            while (i <= Length)
            {
                _data[i] += value;
                i += i & -i;    // LSBの加算
            }
        }

        /// <summary>
        /// [0, <c>end</c>)の部分和を返します。
        /// </summary>
        /// <param name="end">部分和を求める半開区間の終了インデックス</param>
        /// <returns>区間の部分和</returns>
        public long Sum(Index end)
        {
            var i = end.GetOffset(Length);  // 0-indexedの半開区間＝1-indexedの閉区間なので+1は不要
            unchecked
            {
                if ((uint)i >= (uint)_data.Length)
                {
                    throw new ArgumentOutOfRangeException(nameof(end));
                }
            }

            long sum = 0;
            while (i > 0)
            {
                sum += _data[i];
                i -= i & -i;    // LSBの減算
            }
            return sum;
        }

        /// <summary>
        /// <c>range</c>の部分和を返します。
        /// </summary>
        /// <param name="range">部分和を求める半開区間</param>
        /// <returns>区間の部分和</returns>
        public long Sum(Range range) => Sum(range.End) - Sum(range.Start);

        /// <summary>
        /// [<c>start</c>, <c>end</c>)の部分和を返します。
        /// </summary>
        /// <param name="start">部分和を求める半開区間の開始インデックス</param>
        /// <param name="end">部分和を求める半開区間の終了インデックス</param>
        /// <returns>区間の部分和</returns>
        public long Sum(int start, int end) => Sum(end) - Sum(start);

        /// <summary>
        /// [0, <c>index</c>)の部分和が<c>sum</c>未満になる最大の<c>index</c>を返します。
        /// BIT上の各要素は0以上の数である必要があります。
        /// </summary>
        /// <param name="sum"></param>
        /// <returns></returns>
        public int GetLowerBound(long sum)
        {
            int index = 0;
            for (int offset = GetMostSignificantBitOf(Length); offset > 0; offset >>= 1)
            {
                if (index + offset < _data.Length && _data[index + offset] < sum)
                {
                    index += offset;
                    sum -= _data[index];
                }
            }

            return index;

            int GetMostSignificantBitOf(int n)
            {
                int k = 1;
                while ((k << 1) <= n)
                {
                    k <<= 1;
                };
                return k;
            }
        }
    }

    public class BinaryIndexedTree2D
    {
        long[,] _data;
        public int Height { get; }
        public int Width { get; }

        public BinaryIndexedTree2D(int height, int width)
        {
            Height = height;
            Width = width;
            _data = new long[height + 1, width + 1];
        }

        public long this[Index row, Index column]
        {
            get => Sum(row..(row.GetOffset(Height) + 1), column..(column.GetOffset(Width) + 1));
            set
            {
                if (value < 0)
                {
                    throw new ArgumentOutOfRangeException(nameof(value), $"{nameof(value)}は0以上の値である必要があります。");
                }
                AddAt(row, column, value - this[row, column]);
            }
        }

        /// <summary>
        /// 2次元BITの[<c>row</c>, <c>column</c>]に<c>value</c>を足します。
        /// </summary>
        /// <param name="row">加算する行（0-indexed）</param>
        /// <param name="column">加算する列（0-indexed）</param>
        /// <param name="value">加算する値</param>
        public void AddAt(Index row, Index column, long value)
        {
            var initI = row.GetOffset(Height);
            var initJ = column.GetOffset(Width);
            unchecked
            {
                if ((ulong)initI >= (ulong)Height)
                {
                    throw new ArgumentOutOfRangeException(nameof(row));
                }
                if ((ulong)initJ >= (ulong)Width)
                {
                    throw new ArgumentOutOfRangeException(nameof(column));
                }
            }

            initI++;    // 1-indexed
            initJ++;

            for (int i = initI; i <= Height; i += i & -i)
            {
                for (int j = initJ; j <= Width; j += j & -j)
                {
                    _data[i, j] += value;
                }
            }
        }

        /// <summary>
        /// 指定した半開区間の部分和を返します。
        /// </summary>
        /// <param name="row"></param>
        /// <param name="column"></param>
        /// <returns></returns>
        public long Sum(Index row, Index column)
        {
            long sum = 0;
            var initI = row.GetOffset(Height);
            var initJ = column.GetOffset(Width);
            unchecked
            {
                if ((ulong)initI >= (ulong)(Height + 1))
                {
                    throw new ArgumentOutOfRangeException(nameof(row));
                }
                if ((ulong)initJ >= (ulong)(Width + 1))
                {
                    throw new ArgumentOutOfRangeException(nameof(column));
                }
            }

            for (int i = initI; i > 0; i -= i & -i)
            {
                for (int j = initJ; j > 0; j -= j & -j)
                {
                    sum += _data[i, j];
                }
            }
            return sum;
        }

        /// <summary>
        /// 指定した半開区間の部分和を返します。
        /// </summary>
        /// <param name="rows"></param>
        /// <param name="columns"></param>
        /// <returns></returns>
        public long Sum(Range rows, Range columns) => Sum(rows.End, columns.End) - Sum(rows.Start, columns.End) - Sum(rows.End, columns.Start) + Sum(rows.Start, columns.Start);

        /// <summary>
        /// 指定した半開区間の部分和を返します。
        /// </summary>
        /// <param name="beginRow"></param>
        /// <param name="endRow"></param>
        /// <param name="beginColumn"></param>
        /// <param name="endColumn"></param>
        /// <returns></returns>
        public long Sum(int beginRow, int endRow, int beginColumn, int endColumn) => Sum(beginRow..endRow, beginColumn..endColumn);
    }

    public class Counter<T> : IEnumerable<(T key, long count)> where T : IEquatable<T>
    {
        private Dictionary<T, long> _innerDictionary;

        public Counter()
        {
            _innerDictionary = new Dictionary<T, long>();
        }

        public IEnumerator<(T key, long count)> GetEnumerator()
        {
            foreach (var pair in _innerDictionary)
            {
                yield return (key: pair.Key, count: pair.Value);
            }
        }

        System.Collections.IEnumerator System.Collections.IEnumerable.GetEnumerator() => GetEnumerator();

        public long this[T key]
        {
            get
            {
                _innerDictionary.TryGetValue(key, out var count);
                return count;
            }
            set
            {
                _innerDictionary[key] = value;
            }
        }
    }

    public readonly struct BitSet : IEquatable<BitSet>
    {
        readonly uint _value;

        public BitSet(uint value)
        {
            _value = value;
        }
        public bool this[int digit]
        {
            get => ((_value >> digit) & 1) > 0;
        }
        public bool Any => _value > 0;
        public bool None => _value == 0;
        public BitSet SetAt(int digit, bool value) => value ? new BitSet(_value | (1u << digit)) : new BitSet(_value & ~(1u << digit));
        public BitSet Lsb() { unchecked { return new BitSet(_value & (uint)-(int)_value); } }
        public BitSet Reverse()
        {
            unchecked
            {
                uint v = _value;
                v = (v & 0x55555555) << 1 | (v >> 1 & 0x55555555);
                v = (v & 0x33333333) << 2 | (v >> 2 & 0x33333333);
                v = (v & 0x0f0f0f0f) << 4 | (v >> 4 & 0x0f0f0f0f);
                v = (v & 0x00ff00ff) << 8 | (v >> 8 & 0x00ff00ff);
                v = (v & 0x0000ffff) << 16 | (v >> 16 & 0x0000ffff);
                return new BitSet(v);
            }
        }
        public int Count()
        {
            unchecked
            {
                // Hardware Intrinsics未使用
                uint v = _value;
                v = (v & 0x55555555) + (v >> 1 & 0x55555555);
                v = (v & 0x33333333) + (v >> 2 & 0x33333333);
                v = (v & 0x0f0f0f0f) + (v >> 4 & 0x0f0f0f0f);
                v = (v & 0x00ff00ff) + (v >> 8 & 0x00ff00ff);
                v = (v & 0x0000ffff) + (v >> 16 & 0x0000ffff);
                return (int)v;
            }
        }

        public static BitSet Zero => new BitSet(0);
        public static BitSet One => new BitSet(1);
        public static BitSet All => new BitSet(~0u);
        public static BitSet At(int digit) => new BitSet(1u << digit);
        public static BitSet CreateMask(int digit) => new BitSet((1u << digit) - 1);
        public static BitSet operator ++(BitSet bitSet) => new BitSet(bitSet._value + 1);
        public static BitSet operator --(BitSet bitSet) => new BitSet(bitSet._value - 1);
        public static BitSet operator ~(BitSet bitSet) => new BitSet(~bitSet._value);
        public static BitSet operator &(BitSet left, BitSet right) => new BitSet(left._value & right._value);
        public static BitSet operator |(BitSet left, BitSet right) => new BitSet(left._value | right._value);
        public static BitSet operator ^(BitSet left, BitSet right) => new BitSet(left._value ^ right._value);
        public static BitSet operator <<(BitSet bitSet, int n) => new BitSet(bitSet._value << n);
        public static BitSet operator >>(BitSet bitSet, int n) => new BitSet(bitSet._value >> n);
        public static bool operator <(BitSet left, BitSet right) => left._value < right._value;
        public static bool operator <=(BitSet left, BitSet right) => left._value <= right._value;
        public static bool operator >(BitSet left, BitSet right) => left._value > right._value;
        public static bool operator >=(BitSet left, BitSet right) => left._value >= right._value;
        public static bool operator ==(BitSet left, BitSet right) => left.Equals(right);
        public static bool operator !=(BitSet left, BitSet right) => !(left == right);
        public static implicit operator uint(BitSet bitSet) => bitSet._value;

        public override bool Equals(object obj) => obj is BitSet set && Equals(set);
        public bool Equals(BitSet other) => _value == other._value;
        public override string ToString() => Convert.ToString(_value, 2);
        public override int GetHashCode() => _value.GetHashCode();
    }

    public static class SearchExtensions
    {
        class LowerBoundComparer<T> : IComparer<T> where T : IComparable<T>
        {
            public int Compare(T x, T y) => 0 <= x.CompareTo(y) ? 1 : -1;
        }

        class UpperBoundComparer<T> : IComparer<T> where T : IComparable<T>
        {
            public int Compare(T x, T y) => 0 < x.CompareTo(y) ? 1 : -1;
        }

        // https://trsing.hatenablog.com/entry/2019/08/27/211038
        public static int GetGreaterEqualIndex<T>(this ReadOnlySpan<T> span, T inclusiveMin) where T : IComparable<T> => ~span.BinarySearch(inclusiveMin, new UpperBoundComparer<T>());
        public static int GetGreaterThanIndex<T>(this ReadOnlySpan<T> span, T exclusiveMin) where T : IComparable<T> => ~span.BinarySearch(exclusiveMin, new LowerBoundComparer<T>());
        public static int GetLessEqualIndex<T>(this ReadOnlySpan<T> span, T inclusiveMax) where T : IComparable<T> => ~span.BinarySearch(inclusiveMax, new LowerBoundComparer<T>()) - 1;
        public static int GetLessThanIndex<T>(this ReadOnlySpan<T> span, T exclusiveMax) where T : IComparable<T> => ~span.BinarySearch(exclusiveMax, new UpperBoundComparer<T>()) - 1;
        public static int GetGreaterEqualIndex<T>(this Span<T> span, T inclusiveMin) where T : IComparable<T> => ((ReadOnlySpan<T>)span).GetGreaterEqualIndex(inclusiveMin);
        public static int GetGreaterThanIndex<T>(this Span<T> span, T exclusiveMin) where T : IComparable<T> => ((ReadOnlySpan<T>)span).GetGreaterThanIndex(exclusiveMin);
        public static int GetLessEqualIndex<T>(this Span<T> span, T inclusiveMax) where T : IComparable<T> => ((ReadOnlySpan<T>)span).GetLessEqualIndex(inclusiveMax);
        public static int GetLessThanIndex<T>(this Span<T> span, T exclusiveMax) where T : IComparable<T> => ((ReadOnlySpan<T>)span).GetLessThanIndex(exclusiveMax);

        public static int BoundaryBinarySearch(Predicate<int> predicate, int ok, int ng)
        {
            // めぐる式二分探索
            while (Math.Abs(ok - ng) > 1)
            {
                int mid = (ok + ng) / 2;
                if (predicate(mid))
                {
                    ok = mid;
                }
                else
                {
                    ng = mid;
                }
            }
            return ok;
        }

        public static long BoundaryBinarySearch(Predicate<long> predicate, long ok, long ng)
        {
            while (Math.Abs(ok - ng) > 1)
            {
                long mid = (ok + ng) / 2;
                if (predicate(mid))
                {
                    ok = mid;
                }
                else
                {
                    ng = mid;
                }
            }
            return ok;
        }

        public static double Bisection(Func<double, double> f, double a, double b, double eps = 1e-9)
        {
            if (f(a) * f(b) >= 0)
            {
                throw new ArgumentException("f(a)とf(b)は異符号である必要があります。");
            }

            const int maxLoop = 100;
            double mid = (a + b) / 2;

            for (int i = 0; i < maxLoop; i++)
            {
                if (f(a) * f(mid) < 0)
                {
                    b = mid;
                }
                else
                {
                    a = mid;
                }
                mid = (a + b) / 2;
                if (Math.Abs(b - a) < eps)
                {
                    break;
                }
            }
            return mid;
        }
    }

    public static class PermutationAlgorithms
    {
        public static IEnumerable<ReadOnlyMemory<T>> GetPermutations<T>(IEnumerable<T> collection) where T : IComparable<T> => GetPermutations(collection, false);

        public static IEnumerable<ReadOnlyMemory<T>> GetPermutations<T>(IEnumerable<T> collection, bool isSorted) where T : IComparable<T>
        {
            var a = collection.ToArray();

            if (!isSorted && a.Length > 1)
            {
                Array.Sort(a);
            }

            yield return a; // ソート済み初期配列

            if (a.Length <= 2)
            {
                if (a.Length == 2 && a[0].CompareTo(a[1]) != 0)
                {
                    (a[0], a[1]) = (a[1], a[0]);
                    yield return a;
                    yield break;
                }

                yield break;
            }

            bool flag = true;
            while (flag)
            {
                flag = false;
                for (int i = a.Length - 2; i >= 0; i--)
                {
                    // iよりi+1の方が大きい（昇順）なら
                    if (a[i].CompareTo(a[i + 1]) < 0)
                    {
                        // 後ろから見ていってi<jとなるところを探して
                        int j;
                        for (j = a.Length - 1; a[i].CompareTo(a[j]) >= 0; j--) { }

                        // iとjを入れ替えて
                        (a[i], a[j]) = (a[j], a[i]);

                        // i+1以降を反転
                        if (i < a.Length - 2)
                        {
                            var sliced = a.AsSpan().Slice(i + 1);
                            sliced.Reverse();
                        }

                        flag = true;
                        yield return a;
                        break;
                    }
                }
            }
        }
    }

    public static class EnumerableExtensions
    {
        public static IEnumerable<(T1 v1, T2 v2)> Zip<T1, T2>(this (IEnumerable<T1> First, IEnumerable<T2> Second) t)
            => t.First.Zip(t.Second, (v1, v2) => (v1, v2));

        public static IEnumerable<(T1 v1, T2 v2, T3 v3)> Zip<T1, T2, T3>(this (IEnumerable<T1> First, IEnumerable<T2> Second, IEnumerable<T3> Third) t)
            => (t.First, t.Second).Zip().Zip(t.Third, (v12, v3) => (v12.v1, v12.v2, v3));

        public static IEnumerable<(T item, int index)> WithIndex<T>(this IEnumerable<T> source) => source.Select((item, index) => (item, index));
    }

    public static class ArrayExtensions
    {
        public static T[] Fill<T>(this T[] array, T value)
        {
            for (int i = 0; i < array.Length; i++)
                array[i] = value;
            return array;
        }

        public static T[,] Fill<T>(this T[,] array, T value)
        {
            var length0 = array.GetLength(0);
            var length1 = array.GetLength(1);
            for (int i = 0; i < length0; i++)
                for (int j = 0; j < length1; j++)
                    array[i, j] = value;
            return array;
        }

        public static T[,,] Fill<T>(this T[,,] array, T value)
        {
            var length0 = array.GetLength(0);
            var length1 = array.GetLength(1);
            var length2 = array.GetLength(2);
            for (int i = 0; i < length0; i++)
                for (int j = 0; j < length1; j++)
                    for (int k = 0; k < length2; k++)
                        array[i, j, k] = value;
            return array;
        }

        public static T[,,,] Fill<T>(this T[,,,] array, T value)
        {
            var length0 = array.GetLength(0);
            var length1 = array.GetLength(1);
            var length2 = array.GetLength(2);
            var length3 = array.GetLength(3);
            for (int i = 0; i < length0; i++)
                for (int j = 0; j < length1; j++)
                    for (int k = 0; k < length2; k++)
                        for (int l = 0; l < length3; l++)
                            array[i, j, k, l] = value;
            return array;
        }
    }
}

#endregion

#region Graphs

namespace Kujikatsu102.Graphs
{
    public interface INode
    {
        public int Index { get; }
    }

    public interface IEdge<TNode> where TNode : INode
    {
        TNode From { get; }
        TNode To { get; }
    }

    public interface IWeightedEdge<TNode> : IEdge<TNode> where TNode : INode
    {
        long Weight { get; }
    }

    public interface IGraph<TNode, TEdge> where TEdge : IEdge<TNode> where TNode : INode
    {
        IEnumerable<TEdge> this[TNode node] { get; }
        IEnumerable<TEdge> Edges { get; }
        IEnumerable<TNode> Nodes { get; }
        int NodeCount { get; }
    }

    public interface IWeightedGraph<TNode, TEdge> : IGraph<TNode, TEdge> where TEdge : IWeightedEdge<TNode> where TNode : INode { }

    [StructLayout(LayoutKind.Auto)]
    public readonly struct BasicNode : INode, IEquatable<BasicNode>
    {
        public int Index { get; }

        public BasicNode(int index)
        {
            Index = index;
        }

        public override string ToString() => Index.ToString();
        public override bool Equals(object obj) => obj is BasicNode node && Equals(node);
        public bool Equals(BasicNode other) => Index == other.Index;
        public override int GetHashCode() => HashCode.Combine(Index);
        public static bool operator ==(BasicNode left, BasicNode right) => left.Equals(right);
        public static bool operator !=(BasicNode left, BasicNode right) => !(left == right);
        public static implicit operator BasicNode(int value) => new BasicNode(value);
    }

    [StructLayout(LayoutKind.Auto)]
    public readonly struct BasicEdge : IEdge<BasicNode>
    {
        public BasicNode From { get; }
        public BasicNode To { get; }

        public BasicEdge(int from, int to)
        {
            From = from;
            To = to;
        }

        public override string ToString() => $"{From}-->{To}";
    }

    [StructLayout(LayoutKind.Auto)]
    public readonly struct WeightedEdge : IWeightedEdge<BasicNode>
    {
        public BasicNode From { get; }
        public BasicNode To { get; }
        public long Weight { get; }

        public WeightedEdge(int from, int to) : this(from, to, 1) { }

        public WeightedEdge(int from, int to, long weight)
        {
            From = from;
            To = to;
            Weight = weight;
        }

        public override string ToString() => $"{From}--[{Weight}]-->{To}";
    }

    [StructLayout(LayoutKind.Auto)]
    public readonly struct GridNode : INode, IEquatable<GridNode>
    {
        public int Row { get; }
        public int Column { get; }
        public int Index { get; }

        public GridNode(int row, int column, int width)
        {
            Row = row;
            Column = column;
            Index = row * width + column;
        }

        public override string ToString() => $"({Row}, {Column})";
        public override int GetHashCode() => HashCode.Combine(Row, Column, Index);
        public override bool Equals(object obj) => obj is GridNode node && Equals(node);
        public bool Equals(GridNode other) => Row == other.Row && Column == other.Column && Index == other.Index;
        public void Deconstruct(out int row, out int column) { row = Row; column = Column; }
        public static bool operator ==(GridNode left, GridNode right) => left.Equals(right);
        public static bool operator !=(GridNode left, GridNode right) => !(left == right);
    }

    [StructLayout(LayoutKind.Auto)]
    public readonly struct GridEdge : IEdge<GridNode>
    {
        public GridNode From { get; }
        public GridNode To { get; }

        public GridEdge(GridNode from, GridNode to)
        {
            From = from;
            To = to;
        }

        public override string ToString() => $"({From.Row}, {From.Column})-->({To.Row}, {To.Column})";
    }

    public class BasicGraph : IGraph<BasicNode, BasicEdge>
    {
        private readonly List<BasicEdge>[] _edges;
        public IEnumerable<BasicEdge> this[BasicNode node] => _edges[node.Index];
        public IEnumerable<BasicEdge> Edges => Nodes.SelectMany(node => this[node]);
        public IEnumerable<BasicNode> Nodes => Enumerable.Range(0, NodeCount).Select(i => new BasicNode(i));
        public int NodeCount { get; }

        public BasicGraph(int nodeCount) : this(nodeCount, Enumerable.Empty<BasicEdge>()) { }

        public BasicGraph(int nodeCount, IEnumerable<BasicEdge> edges)
        {
            _edges = Enumerable.Repeat(0, nodeCount).Select(_ => new List<BasicEdge>()).ToArray();
            NodeCount = nodeCount;
            foreach (var edge in edges)
            {
                AddEdge(edge);
            }
        }

        public BasicGraph(int nodeCount, IEnumerable<IEnumerable<int>> distances)
        {
            _edges = new List<BasicEdge>[nodeCount];

            int i = 0;
            foreach (var row in distances)
            {
                _edges[i] = new List<BasicEdge>(nodeCount);
                int j = 0;
                foreach (var distance in row)
                {
                    if (distance == 1)
                    {
                        _edges[i].Add(new BasicEdge(i, j++));
                    }
                }
                i++;
            }
        }

        public void AddEdge(BasicEdge edge) => _edges[edge.From.Index].Add(edge);
    }

    public class WeightedGraph : IGraph<BasicNode, WeightedEdge>
    {
        private readonly List<WeightedEdge>[] _edges;
        public IEnumerable<WeightedEdge> this[BasicNode node] => _edges[node.Index];
        public IEnumerable<WeightedEdge> Edges => Nodes.SelectMany(node => this[node]);
        public IEnumerable<BasicNode> Nodes => Enumerable.Range(0, NodeCount).Select(i => new BasicNode(i));
        public int NodeCount { get; }

        public WeightedGraph(int nodeCount) : this(nodeCount, Enumerable.Empty<WeightedEdge>()) { }

        public WeightedGraph(int nodeCount, IEnumerable<WeightedEdge> edges)
        {
            _edges = Enumerable.Repeat(0, nodeCount).Select(_ => new List<WeightedEdge>()).ToArray();
            NodeCount = nodeCount;
            foreach (var edge in edges)
            {
                AddEdge(edge);
            }
        }

        public WeightedGraph(int nodeCount, IEnumerable<IEnumerable<int>> distances)
        {
            _edges = new List<WeightedEdge>[nodeCount];

            int i = 0;
            foreach (var row in distances)
            {
                _edges[i] = new List<WeightedEdge>(nodeCount);
                int j = 0;
                foreach (var distance in row)
                {
                    _edges[i].Add(new WeightedEdge(i, j++, distance));
                }
                i++;
            }
        }

        public void AddEdge(WeightedEdge edge) => _edges[edge.From.Index].Add(edge);
    }

    public class GridGraph : IGraph<GridNode, GridEdge>
    {
        private readonly IReadOnlyList<(int dx, int dy)> _adjacents;
        public int Height { get; }
        public int Width { get; }
        public int NodeCount => Height * Width;

        public IEnumerable<GridEdge> this[GridNode node]
        {
            get
            {
                foreach (var (dx, dy) in _adjacents)
                {
                    var next = new GridNode(node.Row + dx, node.Column + dy, Width);
                    if (CanEnter(next))
                    {
                        yield return new GridEdge(node, next);
                    }
                }
            }
        }

        public IEnumerable<GridEdge> Edges => Nodes.SelectMany(node => this[node]);
        public IEnumerable<GridNode> Nodes => Enumerable.Range(0, Width).SelectMany(x => Enumerable.Range(0, Height).Select(y => new GridNode(x, y, Width)));

        public GridGraph(int height, int width) : this(height, width, new (int dx, int dy)[] { (-1, 0), (1, 0), (0, -1), (0, 1) }) { }

        public GridGraph(int height, int width, IEnumerable<(int dx, int dy)> adjacents)
        {
            Height = height;
            Width = width;
            _adjacents = adjacents.ToArray();
        }

        protected virtual bool CanEnter(GridNode node)
        {
            unchecked
            {
                return (uint)node.Row < Height && (uint)node.Column < Width;
            }
        }
    }

    namespace Algorithms
    {
        // TGraphは派生クラスでいじりたいことがあるのでジェネリクス。
        // TNode, TEdgeはインターフェースで受け取っても動くが、構造体のDevirtualizationを行うため敢えてジェネリクスにしている（型引数長いのでなんとかしたい……）
        public abstract class BfsBase<TGraph, TNode, TEdge, TResult> where TGraph : IGraph<TNode, TEdge> where TEdge : IEdge<TNode> where TNode : INode
        {
            protected readonly TGraph _graph;
            protected bool _completed;

            protected BfsBase(TGraph graph)
            {
                _graph = graph;
            }

            public TResult Search(TNode startNode)
            {
                var todo = new Queue<TNode>();
                var seen = new bool[_graph.NodeCount];
                var cameFrom = new TNode[_graph.NodeCount];
                _completed = false;
                todo.Enqueue(startNode);
                seen[startNode.Index] = true;
                Initialize(startNode);

                while (todo.Count > 0 && !_completed)
                {
                    var current = todo.Dequeue();
                    var isFirstNode = current.Index == startNode.Index;
                    OnPreordering(current, cameFrom[current.Index], isFirstNode);

                    foreach (var edge in _graph[current])
                    {
                        if (seen[edge.To.Index])
                        {
                            continue;
                        }
                        seen[edge.To.Index] = true;
                        cameFrom[edge.To.Index] = edge.From;
                        todo.Enqueue(edge.To);
                    }
                }

                return GetResult();
            }

            protected abstract void Initialize(TNode startNode);
            protected abstract void OnPreordering(TNode current, TNode previous, bool isFirstNode);
            protected abstract TResult GetResult();
        }

        public abstract class DfsBase<TGraph, TNode, TEdge, TResult> where TGraph : IGraph<TNode, TEdge> where TEdge : IEdge<TNode> where TNode : INode
        {
            protected readonly TGraph _graph;
            protected bool _completed;

            protected DfsBase(TGraph graph)
            {
                _graph = graph;
            }

            public TResult Search(TNode startNode)
            {
                var todo = new Stack<TNode>();
                var seen = new bool[_graph.NodeCount];
                var preorderCompleted = new bool[_graph.NodeCount];
                var cameFrom = new TNode[_graph.NodeCount];
                _completed = false;
                todo.Push(startNode);
                seen[startNode.Index] = true;
                Initialize(startNode);

                while (todo.Count > 0 && !_completed)
                {
                    var current = todo.Peek();
                    var isFirstNode = current.Index == startNode.Index;

                    if (!preorderCompleted[current.Index])
                    {
                        // 行きがけ
                        OnPreordering(current, cameFrom[current.Index], isFirstNode);
                        foreach (var edge in _graph[current])
                        {
                            if (seen[edge.To.Index])
                            {
                                continue;
                            }
                            seen[edge.To.Index] = true;
                            cameFrom[edge.To.Index] = edge.From;
                            todo.Push(edge.To);
                        }
                        preorderCompleted[current.Index] = true;
                    }
                    else
                    {
                        // 帰りがけ
                        OnPostordering(current, cameFrom[current.Index], isFirstNode);
                        _ = todo.Pop();
                    }
                }

                return GetResult();
            }

            protected abstract void Initialize(TNode startNode);
            protected abstract void OnPreordering(TNode current, TNode previous, bool isFirstNode);
            protected abstract void OnPostordering(TNode current, TNode previous, bool isFirstNode);
            protected abstract TResult GetResult();
        }

        // 最短経路問題やるだけの問題でIGraphを派生クラスでいじりたいことはほとんどないので普通にインターフェースで受け取る
        public class Dijkstra<TNode, TEdge> where TEdge : IWeightedEdge<TNode> where TNode : INode
        {
            protected readonly IGraph<TNode, TEdge> _graph;

            public Dijkstra(IGraph<TNode, TEdge> graph)
            {
                _graph = graph;
            }

            public long[] GetDistancesFrom(TNode startNode)
            {
                const long Inf = 1L << 60;
                var distances = Enumerable.Repeat(Inf, _graph.NodeCount).ToArray();
                distances[startNode.Index] = 0;
                var todo = new PriorityQueue<State>(false);
                todo.Enqueue(new State(startNode, 0));

                while (todo.Count > 0)
                {
                    var current = todo.Dequeue();
                    if (current.Distance > distances[current.Node.Index])
                    {
                        continue;
                    }

                    foreach (var edge in _graph[current.Node])
                    {
                        var nextDistance = current.Distance + edge.Weight;
                        if (distances[edge.To.Index] > nextDistance)
                        {
                            distances[edge.To.Index] = nextDistance;
                            todo.Enqueue(new State(edge.To, nextDistance));
                        }
                    }
                }

                return distances;
            }

            private readonly struct State : IComparable<State>
            {
                public TNode Node { get; }
                public long Distance { get; }

                public State(TNode node, long distance)
                {
                    Node = node;
                    Distance = distance;
                }

                public int CompareTo(State other) => Distance.CompareTo(other.Distance);
            }
        }

        public class WarshallFloyd<TNode, TEdge> where TEdge : IWeightedEdge<TNode> where TNode : INode
        {
            protected readonly IGraph<TNode, TEdge> _graph;
            const long Inf = 1L << 60;

            public WarshallFloyd(IGraph<TNode, TEdge> graph)
            {
                _graph = graph;
            }

            public long[,] GetDistances()
            {
                var distances = InitializeDistances();
                for (int k = 0; k < _graph.NodeCount; k++)
                {
                    for (int i = 0; i < _graph.NodeCount; i++)
                    {
                        for (int j = 0; j < _graph.NodeCount; j++)
                        {
                            distances[i, j] = Math.Min(distances[i, j], distances[i, k] + distances[k, j]);
                        }
                    }
                }

                // 一応キレイにしておく
                for (int i = 0; i < _graph.NodeCount; i++)
                {
                    for (int j = 0; j < _graph.NodeCount; j++)
                    {
                        if (distances[i, j] >= Inf)
                        {
                            distances[i, j] = long.MaxValue;
                        }
                    }
                }

                return distances;
            }

            private long[,] InitializeDistances()
            {
                var distances = new long[_graph.NodeCount, _graph.NodeCount];

                for (int i = 0; i < _graph.NodeCount; i++)
                {
                    for (int j = 0; j < _graph.NodeCount; j++)
                    {
                        distances[i, j] = Inf;
                    }
                    distances[i, i] = 0;
                }

                foreach (var node in _graph.Nodes)
                {
                    foreach (var edge in _graph.Edges)
                    {
                        distances[edge.From.Index, edge.To.Index] = edge.Weight;
                    }
                }

                return distances;
            }
        }

        public class BellmanFord<TNode, TEdge> where TEdge : IWeightedEdge<TNode> where TNode : INode
        {
            protected readonly List<TEdge> _edges;
            protected readonly int _nodeCount;

            public BellmanFord(IGraph<TNode, TEdge> graph) : this(graph.Edges, graph.NodeCount) { }

            public BellmanFord(IEnumerable<TEdge> edges, int nodeCount)
            {
                _edges = edges.ToList();
                _nodeCount = nodeCount;
            }

            public (long[] distances, bool[] isNegativeCycle) GetDistancesFrom(TNode startNode)
            {
                const long Inf = long.MaxValue >> 1;
                var distances = Enumerable.Repeat(long.MaxValue, _nodeCount).ToArray();
                var isNegativeCycle = new bool[_nodeCount];
                distances[startNode.Index] = 0;

                for (int i = 1; i <= 2 * _nodeCount; i++)
                {
                    foreach (var edge in _edges)
                    {
                        // そもそも出発点に未到達なら無視
                        if (distances[edge.From.Index] < Inf)
                        {
                            if (i <= _nodeCount)
                            {
                                var newCost = distances[edge.From.Index] + edge.Weight;
                                if (distances[edge.To.Index] > newCost)
                                {
                                    distances[edge.To.Index] = newCost;
                                    // N回目に更新されたやつにチェックを付けて、追加でN回伝播させる
                                    if (i == _nodeCount)
                                    {
                                        isNegativeCycle[edge.To.Index] = true;
                                    }
                                }
                            }
                            else if (isNegativeCycle[edge.From.Index])
                            {
                                isNegativeCycle[edge.To.Index] = true;
                            }
                        }
                    }
                }

                // 一応キレイにしておく
                for (int i = 0; i < _nodeCount; i++)
                {
                    if (isNegativeCycle[i])
                    {
                        distances[i] = long.MinValue;
                    }
                    else if (distances[i] >= Inf)
                    {
                        distances[i] = long.MaxValue;
                    }
                }

                return (distances, isNegativeCycle);
            }
        }

        public interface ITreeDpState<TSet> : IMonoid<TSet> where TSet : ITreeDpState<TSet>, new()
        {
            public TSet AddRoot();
        }

        public class Rerooting<TNode, TEdge, TTreeDpState> where TEdge : IEdge<TNode> where TNode : struct, INode where TTreeDpState : ITreeDpState<TTreeDpState>, new()
        {
            readonly IGraph<TNode, TEdge> _graph;
            readonly TTreeDpState _identity;
            readonly Dictionary<int, TTreeDpState>[] _dp;
            readonly TTreeDpState[] _result;

            public Rerooting(IGraph<TNode, TEdge> graph)
            {
                _graph = graph;
                _identity = new TTreeDpState().Identity;
                _dp = new Dictionary<int, TTreeDpState>[_graph.NodeCount];
                _result = new TTreeDpState[_graph.NodeCount];
            }

            public TTreeDpState[] Solve()
            {
                DepthFirstSearch(_graph.Nodes.First(), null);
                Reroot(_graph.Nodes.First(), null, _identity);
                return _result;
            }

            private TTreeDpState DepthFirstSearch(TNode root, TNode? parent)
            {
                var sum = _identity;
                _dp[root.Index] = new Dictionary<int, TTreeDpState>();

                foreach (var edge in _graph[root])
                {
                    if (edge.To.Index == parent?.Index)
                        continue;
                    _dp[root.Index].Add(edge.To.Index, DepthFirstSearch(edge.To, root));
                    sum = sum.Merge(_dp[root.Index][edge.To.Index]);
                }
                return sum.AddRoot();
            }

            private void Reroot(TNode root, TNode? parent, TTreeDpState toAdd)
            {
                var edges = _graph[root].ToArray();

                foreach (var edge in edges)
                {
                    if (edge.To.Index == parent?.Index)
                    {
                        _dp[root.Index].Add(edge.To.Index, toAdd);
                        break;
                    }
                }

                var dp = GetPrefixSum(root, edges);

                for (int i = 0; i < edges.Length; i++)
                {
                    var child = edges[i].To;
                    if (child.Index == parent?.Index)
                        continue;
                    Reroot(child, root, dp[i].AddRoot());
                }
            }

            private TTreeDpState[] GetPrefixSum(TNode root, TEdge[] edges)
            {
                // 左右からの累積和
                int sumSize = edges.Length + 1;
                var sumLeft = new TTreeDpState[sumSize];
                sumLeft[0] = _identity;
                for (int i = 0; i < edges.Length; i++)
                {
                    var child = edges[i].To;
                    sumLeft[i + 1] = sumLeft[i].Merge(_dp[root.Index][child.Index]);
                }

                var sumRight = new TTreeDpState[sumSize];
                sumRight[^1] = _identity;
                for (int i = edges.Length - 1; i >= 0; i--)
                {
                    var child = edges[i].To;
                    sumRight[i] = sumRight[i + 1].Merge(_dp[root.Index][child.Index]);
                }

                _result[root.Index] = sumLeft[^1].AddRoot();

                // 頂点iを除いた累積
                var dp = new TTreeDpState[edges.Length];
                for (int i = 0; i < dp.Length; i++)
                {
                    dp[i] = sumLeft[i].Merge(sumRight[i + 1]);
                }

                return dp;
            }
        }
    }
}

#endregion