Problem:
Score : 300 points

Problem Statement

You have a pot and N ingredients. Each ingredient has a real number parameter called value, and the value of the i-th ingredient (1 \leq i \leq N) is v_i.

When you put two ingredients in the pot, they will vanish and result in the formation of a new ingredient. The value of the new ingredient will be (x + y) / 2 where x and y are the values of the ingredients consumed, and you can put this ingredient again in the pot.

After you compose ingredients in this way N-1 times, you will end up with one ingredient. Find the maximum possible value of this ingredient.

Constraints

2 \leq N \leq 50

1 \leq v_i \leq 1000

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
v_1 v_2 \ldots v_N

Output

Print a decimal number (or an integer) representing the maximum possible value of the last ingredient remaining.

Your output will be judged correct when its absolute or relative error from the judge's output is at most 10^{-5}.

Sample Input 1

2
3 4

Sample Output 1

3.5

If you start with two ingredients, the only choice is to put both of them in the pot. The value of the ingredient resulting from the ingredients of values 3 and 4 is (3 + 4) / 2 = 3.5.

Printing 3.50001, 3.49999, and so on will also be accepted.

Sample Input 2

3
500 300 200

Sample Output 2

375

You start with three ingredients this time, and you can choose what to use in the first composition. There are three possible choices:

Use the ingredients of values 500 and 300 to produce an ingredient of value (500 + 300) / 2 = 400. The next composition will use this ingredient and the ingredient of value 200, resulting in an ingredient of value (400 + 200) / 2 = 300.

Use the ingredients of values 500 and 200 to produce an ingredient of value (500 + 200) / 2 = 350. The next composition will use this ingredient and the ingredient of value 300, resulting in an ingredient of value (350 + 300) / 2 = 325.

Use the ingredients of values 300 and 200 to produce an ingredient of value (300 + 200) / 2 = 250. The next composition will use this ingredient and the ingredient of value 500, resulting in an ingredient of value (250 + 500) / 2 = 375.

Thus, the maximum possible value of the last ingredient remaining is 375.

Printing 375.0 and so on will also be accepted.

Sample Input 3

5
138 138 138 138 138

Sample Output 3

138

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

using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Text;
using Kujikatsu004.Algorithms;
using Kujikatsu004.Collections;
using Kujikatsu004.Extensions;
using Kujikatsu004.Numerics;
using Kujikatsu004.Questions;

namespace Kujikatsu004.Questions
{
    /// <summary>
    /// https://atcoder.jp/contests/abc138/tasks/abc138_c
    /// </summary>
    public class QuestionB : AtCoderQuestionBase
    {
        public override IEnumerable<object> Solve(TextReader inputStream)
        {
            var _ = inputStream.ReadInt();
            var v = inputStream.ReadIntArray();
            Array.Sort(v);

            double sum = v[0];

            foreach (var vi in v.Skip(1))
            {
                sum = (sum + vi) / 2;
            }

            yield return sum;
        }
    }
}


namespace Kujikatsu004
{
    class Program
    {
        static void Main(string[] args)
        {
            IAtCoderQuestion question = new QuestionB();
            var answers = question.Solve(Console.In);

            var writer = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
            Console.SetOut(writer);
            foreach (var answer in answers)
            {
                Console.WriteLine(answer);
            }
            Console.Out.Flush();
        }
    }
}

#region Base Class

namespace Kujikatsu004.Questions
{

    public interface IAtCoderQuestion
    {
        IEnumerable<object> Solve(string input);
        IEnumerable<object> Solve(TextReader inputStream);
    }

    public abstract class AtCoderQuestionBase : IAtCoderQuestion
    {
        public IEnumerable<object> Solve(string input)
        {
            var stream = new MemoryStream(Encoding.Unicode.GetBytes(input));
            var reader = new StreamReader(stream, Encoding.Unicode);

            return Solve(reader);
        }

        public abstract IEnumerable<object> Solve(TextReader inputStream);
    }
}

#endregion

#region Algorithm

namespace Kujikatsu004.Numerics
{
    public static class NumericalAlgorithms
    {
        public static long Gcd(long a, long b)
        {
            if (a <= 0)
            {
                throw new ArgumentOutOfRangeException(nameof(a), $"{nameof(b)}は正の整数である必要があります。");
            }
            if (b <= 0)
            {
                throw new ArgumentOutOfRangeException(nameof(b), $"{nameof(b)}は正の整数である必要があります。");
            }
            if (a < b)
            {
                (a, b) = (b, a);
            }

            while (b != 0)
            {
                (a, b) = (b, a % b);
            }
            return a;
        }

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

    [System.Runtime.InteropServices.StructLayout(System.Runtime.InteropServices.LayoutKind.Auto)]
    public readonly struct Modular : IEquatable<Modular>, IComparable<Modular>
    {
        private const int _defaultMod = 1000000007;
        public int Value { get; }
        private readonly int? _mod;     // int?にすると遅くなるけど意図せずmod=0とかで初期化されて意味不明な計算結果になるよりはマシ
        public int Mod => _mod ?? throw new InvalidOperationException($"{nameof(Mod)}がnullです。new {nameof(Modular)}(long value, int mod)で初期化してください。");

        public Modular(long value, int mod = _defaultMod)
        {
            if (mod < 2 || mod > 1073741789)
            {
                // 1073741789はint.MaxValue / 2 = 1073741823以下の最大の素数
                throw new ArgumentOutOfRangeException(nameof(mod), $"{nameof(mod)}は2以上1073741789以下の素数でなければなりません。");
            }
            _mod = mod;

            if (value >= 0 && value < mod)
            {
                Value = (int)value;
            }
            else
            {
                Value = (int)(value % mod);
                if (Value < 0)
                {
                    Value += mod;
                }
            }
        }

        public static Modular operator +(in Modular a, in Modular b)
        {
            CheckModEquals(a, b);

            var result = a.Value + b.Value;
            if (result > a.Mod)
            {
                result -= a.Mod;    // 剰余演算を避ける
            }
            return new Modular(result, a.Mod);
        }

        public static Modular operator -(in Modular a, in Modular b)
        {
            CheckModEquals(a, b);

            var result = a.Value - b.Value;
            if (result < 0)
            {
                result += a.Mod;    // 剰余演算を避ける
            }
            return new Modular(result, a.Mod);
        }

        public static Modular operator *(in Modular a, in Modular b)
        {
            CheckModEquals(a, b);
            return new Modular((long)a.Value * b.Value, a.Mod);
        }

        public static Modular operator /(in Modular a, in Modular b)
        {
            CheckModEquals(a, b);
            return a * Pow(b, a.Mod - 2);
        }

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

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

        public static Modular Pow(in Modular a, int n)
        {
            switch (n)
            {
                case 0:
                    return new Modular(1, a.Mod);
                case 1:
                    return a;
                case int m when m >= 0: // ジャンプテーブル化はできなくなる
                    var p = Pow(a, m >> 1);             // m / 2
                    return p * p * Pow(a, m & 0x01);    // m % 2
                default:
                    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 Dictionary<int, List<int>> _factorialCache;
        private static Dictionary<int, List<int>> FactorialCache => _factorialCache ??= new Dictionary<int, List<int>>();
        private static Dictionary<int, int[]> FactorialInverseCache { get; } = new Dictionary<int, int[]>();
        const int maxFactorial = 1000000;

        public static Modular Factorial(int n, int mod = _defaultMod)
        {
            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 (mod < 2)
            {
                throw new ArgumentOutOfRangeException(nameof(mod), $"{nameof(mod)}は2以上の素数でなければなりません。");
            }

            if (!FactorialCache.ContainsKey(mod))
            {
                FactorialCache.Add(mod, new List<int>() { 1 });
            }

            var cache = FactorialCache[mod];
            for (int i = cache.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
            {
                cache.Add((int)((long)cache[i - 1] * i % mod));
            }
            return new Modular(cache[n], mod);
        }

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

        public static Modular Combination(int n, int r, int mod = _defaultMod)
        {
            if (!FactorialInverseCache.ContainsKey(mod))
            {
                InitializeCombinationTable(maxFactorial, mod);
            }
            CheckNR(n, r);
            r = Math.Min(r, n - r);
            return new Modular(FactorialCache[mod][n], mod) * new Modular(FactorialInverseCache[mod][r], mod) * new Modular(FactorialInverseCache[mod][n - r], mod);
        }

        private static void InitializeCombinationTable(int max, int mod)
        {
            Factorial(max);
            FactorialInverseCache.Add(mod, new int[max + 1]);

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

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

        public static Modular[] CreateArray(int length, int mod = _defaultMod) => Enumerable.Repeat(new Modular(0, mod), length).ToArray();

        private static void CheckModEquals(in Modular a, in Modular b)
        {
            if (a.Mod != b.Mod)
            {
                throw new ArgumentException($"{nameof(a)}, {nameof(b)}", $"両者の法{nameof(Mod)}は等しくなければなりません。");
            }
        }

        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} (mod {Mod})";

        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 && Mod == other.Mod;

        public int CompareTo([System.Diagnostics.CodeAnalysis.AllowNull] Modular other)
        {
            CheckModEquals(this, other);
            return Value.CompareTo(other.Value);
        }

        public override int GetHashCode() => (Value, Mod).GetHashCode();
    }

    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 Multiply(TSet other);
        public static TSet operator *(ISemigroup<TSet> a, TSet b) => a.Multiply(b);
    }

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

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

}

namespace Kujikatsu004.Algorithms
{
    public static class StringAlgorithms
    {
        public static int[] ZAlgorithm(string s) => ZAlgorithm(s.AsSpan());

        public static int[] ZAlgorithm(ReadOnlySpan<char> s)
        {
            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] == 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;
        }
    }

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

        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();
    }

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

    public class Rerooting<TTreeDpState> where TTreeDpState : ITreeDpState<TTreeDpState>, new()
    {
        readonly IReadOnlyList<int>[] _graph;
        readonly TTreeDpState _identity;
        readonly Dictionary<int, TTreeDpState>[] _dp;
        readonly TTreeDpState[] _result;

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

        public TTreeDpState[] Solve()
        {
            DepthFirstSearch();
            Reroot();
            return _result;
        }

        private TTreeDpState DepthFirstSearch() => DepthFirstSearch(0, -1);

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

            foreach (var child in _graph[root])
            {
                if (child == parent)
                    continue;
                _dp[root].Add(child, DepthFirstSearch(child, root));
                sum *= _dp[root][child];
            }
            return sum.AddRoot();
        }

        private void Reroot() => Reroot(0, -1, _identity);

        private void Reroot(int root, int parent, TTreeDpState toAdd)
        {
            var degree = _graph[root].Count;
            for (int i = 0; i < _graph[root].Count; i++)
            {
                var child = _graph[root][i];
                if (child == parent)
                {
                    _dp[root].Add(child, toAdd);
                    break;
                }
            }

            // 累積和
            int sumSize = degree + 1;
            var sumLeft = new TTreeDpState[sumSize];
            sumLeft[0] = _identity;
            for (int i = 0; i < degree; i++)
            {
                var child = _graph[root][i];
                sumLeft[i + 1] = sumLeft[i] * _dp[root][child];
            }

            var sumRight = new TTreeDpState[sumSize];
            sumRight[degree] = _identity;
            for (int i = degree - 1; i >= 0; i--)
            {
                var child = _graph[root][i];
                sumRight[i] = sumRight[i + 1] * _dp[root][child];
            }
            _result[root] = sumLeft[degree].AddRoot();

            for (int i = 0; i < _graph[root].Count; i++)
            {
                var child = _graph[root][i];
                if (child == parent)
                    continue;
                var dp = sumLeft[i] * sumRight[i + 1];
                Reroot(child, root, dp.AddRoot());
            }
        }
    }
}

#endregion

#region Collections

namespace Kujikatsu004.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 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[_leafOffset..(_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] * _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 * 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] * _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 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[] SetAll<T>(this T[] array, Func<int, T> func)
        {
            for (int i = 0; i < array.Length; i++)
                array[i] = func(i);
            return array;
        }

        public static T[,] SetAll<T>(this T[,] array, Func<int, int, T> func)
        {
            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] = func(i, j);
            return array;
        }

        public static T[,,] SetAll<T>(this T[,,] array, Func<int, int, int, T> func)
        {
            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] = func(i, j, k);
            return array;
        }

        public static T[,,,] SetAll<T>(this T[,,,] array, Func<int, int, int, int, T> func)
        {
            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] = func(i, j, k, l);
            return array;
        }
    }
}

#endregion

#region Graphs

namespace Kujikatsu004.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
    {
        int 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 { }

    [System.Runtime.InteropServices.StructLayout(System.Runtime.InteropServices.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);
    }

    [System.Runtime.InteropServices.StructLayout(System.Runtime.InteropServices.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}";
    }

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

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

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

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

    [System.Runtime.InteropServices.StructLayout(System.Runtime.InteropServices.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);
    }

    [System.Runtime.InteropServices.StructLayout(System.Runtime.InteropServices.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);
            }
        }
    }
}

#endregion

#region Extensions

namespace Kujikatsu004.Extensions
{
    public static class TextReaderExtensions
    {
        public static int ReadInt(this TextReader reader) => int.Parse(ReadString(reader));
        public static long ReadLong(this TextReader reader) => long.Parse(ReadString(reader));
        public static double ReadDouble(this TextReader reader) => double.Parse(ReadString(reader));
        public static string ReadString(this TextReader reader) => reader.ReadLine();

        public static int[] ReadIntArray(this TextReader reader, char separator = ' ') => ReadStringArray(reader, separator).Select(int.Parse).ToArray();
        public static long[] ReadLongArray(this TextReader reader, char separator = ' ') => ReadStringArray(reader, separator).Select(long.Parse).ToArray();
        public static double[] ReadDoubleArray(this TextReader reader, char separator = ' ') => ReadStringArray(reader, separator).Select(double.Parse).ToArray();
        public static string[] ReadStringArray(this TextReader reader, char separator = ' ') => reader.ReadLine().Split(separator);

        // Supports primitive type only.
        public static T1 ReadValue<T1>(this TextReader reader) => (T1)Convert.ChangeType(reader.ReadLine(), typeof(T1));

        public static (T1, T2) ReadValue<T1, T2>(this TextReader reader, char separator = ' ')
        {
            var inputs = ReadStringArray(reader, separator);
            var v1 = (T1)Convert.ChangeType(inputs[0], typeof(T1));
            var v2 = (T2)Convert.ChangeType(inputs[1], typeof(T2));
            return (v1, v2);
        }

        public static (T1, T2, T3) ReadValue<T1, T2, T3>(this TextReader reader, char separator = ' ')
        {
            var inputs = ReadStringArray(reader, separator);
            var v1 = (T1)Convert.ChangeType(inputs[0], typeof(T1));
            var v2 = (T2)Convert.ChangeType(inputs[1], typeof(T2));
            var v3 = (T3)Convert.ChangeType(inputs[2], typeof(T3));
            return (v1, v2, v3);
        }

        public static (T1, T2, T3, T4) ReadValue<T1, T2, T3, T4>(this TextReader reader, char separator = ' ')
        {
            var inputs = ReadStringArray(reader, separator);
            var v1 = (T1)Convert.ChangeType(inputs[0], typeof(T1));
            var v2 = (T2)Convert.ChangeType(inputs[1], typeof(T2));
            var v3 = (T3)Convert.ChangeType(inputs[2], typeof(T3));
            var v4 = (T4)Convert.ChangeType(inputs[3], typeof(T4));
            return (v1, v2, v3, v4);
        }

        public static (T1, T2, T3, T4, T5) ReadValue<T1, T2, T3, T4, T5>(this TextReader reader, char separator = ' ')
        {
            var inputs = ReadStringArray(reader, separator);
            var v1 = (T1)Convert.ChangeType(inputs[0], typeof(T1));
            var v2 = (T2)Convert.ChangeType(inputs[1], typeof(T2));
            var v3 = (T3)Convert.ChangeType(inputs[2], typeof(T3));
            var v4 = (T4)Convert.ChangeType(inputs[3], typeof(T4));
            var v5 = (T5)Convert.ChangeType(inputs[4], typeof(T5));
            return (v1, v2, v3, v4, v5);
        }

        public static (T1, T2, T3, T4, T5, T6) ReadValue<T1, T2, T3, T4, T5, T6>(this TextReader reader, char separator = ' ')
        {
            var inputs = ReadStringArray(reader, separator);
            var v1 = (T1)Convert.ChangeType(inputs[0], typeof(T1));
            var v2 = (T2)Convert.ChangeType(inputs[1], typeof(T2));
            var v3 = (T3)Convert.ChangeType(inputs[2], typeof(T3));
            var v4 = (T4)Convert.ChangeType(inputs[3], typeof(T4));
            var v5 = (T5)Convert.ChangeType(inputs[4], typeof(T5));
            var v6 = (T6)Convert.ChangeType(inputs[5], typeof(T6));
            return (v1, v2, v3, v4, v5, v6);
        }

        public static (T1, T2, T3, T4, T5, T6, T7) ReadValue<T1, T2, T3, T4, T5, T6, T7>(this TextReader reader, char separator = ' ')
        {
            var inputs = ReadStringArray(reader, separator);
            var v1 = (T1)Convert.ChangeType(inputs[0], typeof(T1));
            var v2 = (T2)Convert.ChangeType(inputs[1], typeof(T2));
            var v3 = (T3)Convert.ChangeType(inputs[2], typeof(T3));
            var v4 = (T4)Convert.ChangeType(inputs[3], typeof(T4));
            var v5 = (T5)Convert.ChangeType(inputs[4], typeof(T5));
            var v6 = (T6)Convert.ChangeType(inputs[5], typeof(T6));
            var v7 = (T7)Convert.ChangeType(inputs[6], typeof(T7));
            return (v1, v2, v3, v4, v5, v6, v7);
        }

        public static (T1, T2, T3, T4, T5, T6, T7, T8) ReadValue<T1, T2, T3, T4, T5, T6, T7, T8>(this TextReader reader, char separator = ' ')
        {
            var inputs = ReadStringArray(reader, separator);
            var v1 = (T1)Convert.ChangeType(inputs[0], typeof(T1));
            var v2 = (T2)Convert.ChangeType(inputs[1], typeof(T2));
            var v3 = (T3)Convert.ChangeType(inputs[2], typeof(T3));
            var v4 = (T4)Convert.ChangeType(inputs[3], typeof(T4));
            var v5 = (T5)Convert.ChangeType(inputs[4], typeof(T5));
            var v6 = (T6)Convert.ChangeType(inputs[5], typeof(T6));
            var v7 = (T7)Convert.ChangeType(inputs[6], typeof(T7));
            var v8 = (T8)Convert.ChangeType(inputs[7], typeof(T8));
            return (v1, v2, v3, v4, v5, v6, v7, v8);
        }
    }
}

#endregion