using System.ComponentModel;
using System.Diagnostics.CodeAnalysis;
using System.Numerics;
using System.Runtime.CompilerServices;
using System.Runtime.InteropServices;
using System.Collections;
using System.Text;
using System.Text.RegularExpressions;
using static Tmpl;
using static CPDebug;
using System.Collections.Specialized;
using System.Globalization;
using System.Diagnostics;

StreamWriter writer = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
//using StreamWriter writer = new StreamWriter(File.Open("turn.txt", FileMode.Create, FileAccess.ReadWrite));
Console.SetOut(writer);
//using StreamReader reader = new StreamReader(File.Open("in.txt", FileMode.Open));
//Console.SetIn(reader);

Solver.Solve();

Console.Out.Flush();

public static class Solver
{
    private static readonly AtCoderIO cin = new AtCoderIO();

    public static unsafe void Solve()
    {
        int N = cin.Int();
        int F = cin.Int();
        int[] A = cin.IntArray(N);
        int[] B = cin.IntArray(N);
        int[] C = cin.IntArray(N);

        int max = F * N;

        bool[] dp = new bool[max + 1];
        bool[] nextdp = new bool[max + 1];
        dp[A[0]] = true;
        dp[B[0]] = true;
        dp[C[0]] = true;
        int one = 0;
        for (int i = 0; i < max; i++)
        {
            if (dp[i]) one++;
        }
        print(one);

        for (int i = 2; i <= N; i++)
        {
            int ans = 0;
            for (int j = 0; j <= max; j++)
            {
                if (j >= A[i - 1])
                {
                    nextdp[j] |= dp[j - A[i - 1]];
                }
                if (j >= B[i - 1])
                {
                    nextdp[j] |= dp[j - B[i - 1]];
                }
                if (j >= C[i - 1])
                {
                    nextdp[j] |= dp[j - C[i - 1]];
                }

                if (nextdp[j]) ans++;
            }

            print(ans);
            (dp, nextdp) = (nextdp, dp);
        }
    }
}

public struct Vector<T> where T : struct, INumber<T>
{
    private T[] _data;
    private int _size;

    public T this[int index]
    {
        get => _data[index];
        set => _data[index] = value;
    }

    public int Size => _size;
    public T[] Data => _data;

    public Vector(int size)
    {
        _data = new T[size];
        _size = size;
    }
}

public struct Matrix<T> where T : struct, INumber<T>
{
    private T[,] _data;
    private int _size;

    public T this[int r, int c]
    {
        get
        {
            return _data[r, c];
        }
        set
        {
            _data[r, c] = value;
        }
    }

    public T[,] Data => _data;
    public int Size => _size;

    public Matrix(int size)
    {
        _size = size;
        _data = new T[size, size];
    }

    public static Matrix<T> operator +(Matrix<T> left, Matrix<T> right)
    {
        if (left.Size != right.Size)
        {
            throw new InvalidOperationException();
        }
        Matrix<T> dest = new(left.Size);
        for (int r = 0; r < left.Size; r++)
        {
            for (int c = 0; c < left.Size; c++)
            {
                dest[r, c] = left[r, c] + right[r, c];
            }
        }

        return dest;
    }

    public static Matrix<T> operator -(Matrix<T> left, Matrix<T> right)
    {
        if (left.Size != right.Size)
        {
            throw new InvalidOperationException();
        }
        Matrix<T> dest = new(left.Size);
        for (int r = 0; r < left.Size; r++)
        {
            for (int c = 0; c < left.Size; c++)
            {
                dest[r, c] = left[r, c] - right[r, c];
            }
        }

        return dest;
    }

    public static Matrix<T> operator *(Matrix<T> left, Matrix<T> right)
    {
        if (left.Size != right.Size)
        {
            throw new InvalidOperationException();
        }

        Matrix<T> dest = new(left.Size);

        for (int r = 0; r < dest.Size; r++)
        {
            for (int c = 0; c < dest.Size; c++)
            {
                for (int i = 0; i < dest.Size; i++)
                {
                    dest[r, c] += right[i, c] * left[r, i];
                }
            }
        }

        return dest;
    }

    public override string ToString()
    {
        StringBuilder sb = new();
        for (int r = 0; r < _size; r++)
        {
            sb.Append('|');
            for (int c = 0; c < _size; c++)
            {
                sb.Append(_data[r, c]);
                if (c != _size - 1)
                {
                    sb.Append(',');
                    sb.Append('\t');
                }
                
            }
            sb.Append('|');
            sb.Append(Environment.NewLine);
        }

        return sb.ToString();
    }
}

static class Constants
{
    public const long Mod = 998244353L;
    //public const long Mod = 10007L;
    //public const long Mod = 1000000007L;
}

public readonly struct Point
{
    public readonly int X;
    public readonly int Y;

    public Point(int x, int y)
    {
        X = x;
        Y = y;
    }

    public override int GetHashCode() => (X, Y).GetHashCode();

    public override string ToString()
    {
        return $"({X}, {Y})";
    }
}

public sealed class ReverseComparer<T> : IComparer<T>
{
    public static readonly ReverseComparer<T> Default = new ReverseComparer<T>(Comparer<T>.Default);

    public static ReverseComparer<T> Reverse(IComparer<T> comparer)
    {
        return new ReverseComparer<T>(comparer);
    }

    private readonly IComparer<T> comparer = Default;

    private ReverseComparer(IComparer<T> comparer)
    {
        this.comparer = comparer;
    }

    public int Compare(T x, T y)
    {
        return comparer.Compare(y, x);
    }
}

public sealed class AtCoderIO
{
    Queue<string> _readQueue = new Queue<string>();

    private void LoadQueue()
    {
        if (_readQueue.Count > 0) return;
        string line = Console.ReadLine();
        string[] split = line.Split(' ', StringSplitOptions.RemoveEmptyEntries);
        for (int i = 0; i < split.Length; i++) _readQueue.Enqueue(split[i]);
    }

    private void Guard()
    {
        if (_readQueue.Count == 0)
        {
            throw new Exception("NO DATA TO READ");
        }
    }

    public int Int()
    {
        LoadQueue();
        Guard();
        return int.Parse(_readQueue.Dequeue());
    }

    public long Long()
    {
        LoadQueue();
        Guard();
        return long.Parse(_readQueue.Dequeue());
    }

    public string String()
    {
        LoadQueue();
        Guard();
        return _readQueue.Dequeue();
    }

    public short Short()
    {
        LoadQueue();
        Guard();
        return short.Parse(_readQueue.Dequeue());
    }

    public byte Byte()
    {
        LoadQueue();
        Guard();
        return byte.Parse(_readQueue.Dequeue());
    }

    public char Char()
    {
        LoadQueue();
        Guard();
        return char.Parse(_readQueue.Dequeue());
    }

    public double Double()
    {
        LoadQueue();
        Guard();
        return double.Parse(_readQueue.Dequeue());
    }

    public float Float()
    {
        LoadQueue();
        Guard();
        return float.Parse(_readQueue.Dequeue());
    }

    public ModInt ModInt()
    {
        return new ModInt(Long());
    }

    public T Read<T>()
    {
        Type type = typeof(T);
        if (type == typeof(int)) return (T)(object)Int();
        else if (type == typeof(long)) return (T)(object)Long();
        else if (type == typeof(float)) return (T)(object)Float();
        else if (type == typeof(double)) return (T)(object)Double();
        else if (type == typeof(short)) return (T)(object)Short();
        else if (type == typeof(byte)) return (T)(object)Byte();
        else if (type == typeof(char)) return (T)(object)Char();
        else if (type == typeof(string)) return (T)(object)String();
        else if (type == typeof(ModInt)) return (T)(object)ModInt();
        else return default(T);
    }

    public int[] IntArray(int n)
    {
        if (n == 0) return Array.Empty<int>();

        int[] arr = new int[n];
        for (int i = 0; i < n; i++)
        {
            arr[i] = Int();
        }

        return arr;
    }

    public int[] ZeroIndexedPermutation(int n)
    {
        if (n == 0) return Array.Empty<int>();

        int[] arr = new int[n];
        for (int i = 0; i < n; i++)
        {
            arr[i] = Int() - 1;
        }

        return arr;
    }

    public long[] LongArray(int n)
    {
        if (n == 0) return Array.Empty<long>();

        long[] arr = new long[n];
        for (int i = 0; i < n; i++)
        {
            arr[i] = Long();
        }

        return arr;
    }

    public double[] DoubleArray(int n)
    {
        if (n == 0) return Array.Empty<double>();

        double[] arr = new double[n];
        for (int i = 0; i < n; i++)
        {
            arr[i] = Double();
        }

        return arr;
    }

    public ModInt[] ModIntArray(int n)
    {
        if (n == 0) return Array.Empty<ModInt>();

        ModInt[] arr = new ModInt[n];
        for (int i = 0; i < n; i++)
        {
            arr[i] = (ModInt)Long();
        }

        return arr;
    }

    public T[] ReadArray<T>(int n)
    {
        if (n == 0) return Array.Empty<T>();

        T[] arr = new T[n];
        for (int i = 0; i < n; i++)
        {
            arr[i] = Read<T>();
        }

        return arr;
    }
}

public static class CPDebug
{
    public static void check<T>(T value, [CallerArgumentExpression(nameof(value))] string name = "value")
    {
#if DEBUG
        Console.WriteLine($"[DEBUG] {name}={value.ToString()}");
#endif
    }

    public static void check<T>(IEnumerable<T> value, [CallerArgumentExpression(nameof(value))] string name = "value")
    {
#if DEBUG
        Console.WriteLine($"[DEBUG] {name}=({string.Join(' ', value)})");
#endif
    }
}

public static class Tmpl
{
    public static int combination(int n, int r)
    {
        int c = 1;
        for (int i = 1; i <= r; i++)
        {
            c = c * (n - i + 1) / i;
        }
        return c;
    }

    public static long combination(long n, long r)
    {
        long c = 1;
        for (long i = 1; i <= r; i++)
        {
            c = c * (n - i + 1) / i;
        }
        return c;
    }

    public static long pow(long b, long exp)
    {
        if (exp == 0) return 1;
        if (exp == 1) return b;

        long m = pow(b, exp / 2L);
        m *= m;
        if (exp % 2L == 1) m *= b;

        return m;
    }

    public static long modpow(long b, long exp, long mod)
    {
        if (exp == 0) return 1;
        if (exp == 1) return b % mod;

        long m = modpow(b, exp / 2L, mod);
        m *= m;
		m %= mod;
        if (exp % 2L == 1) m *= b % mod;
        m %= mod;

        return m;
    }

	static long modinv( long a, long mod )
	{
		long b = mod, u = 1, v = 0;
		while ( b > 0 )
		{
			long t = a / b;
			a -= t * b; swap( ref a, ref b );
			u -= t * v; swap( ref u, ref v );
		}
		u %= mod;
		if (u < 0) u += mod;
		return u;
	}

    public static long calcLcm(long a, long b)
    {
        return a * b / calcGcd(a, b);
    }

    public static long calcGcd(long a, long b)
    {
        if (a % b == 0) return b;
        return calcGcd(b, a % b);
    }

    // ax ≡ b (mod m)なる最小のxを求める
    public static bool solveModLinear(long a, long b, long m, out long minSolution)
    {
        long gcd = calcGcd(calcGcd(a, b), m);
        a /= gcd;
        b /= gcd;
        m /= gcd;
        if (calcGcd(a, m) == 1)
        {
            long inv = modinv(a, m);
            minSolution = (b * inv) % m;
            return true;
        }
        minSolution = 0;
        return false;
    }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void YesNo(bool t)
    {
        Console.WriteLine(t ? "Yes" : "No");
    }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void YESNO(bool t)
    {
        Console.WriteLine(t ? "YES" : "NO");
    }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void yesno(bool t)
    {
        Console.WriteLine(t ? "yes" : "no");
    }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static bool checkBit(int bit, int n)
    {
        return ((1 << n) & bit) != 0;
    }

    public static bool nextCombination(int n, int r, int[] array)
    {
        int i = array.Length - 1;
        while (i >= 0 && array[i] == i + n - r)
        {
            i--;
        }

        if (i < 0) return false;

        array[i]++;
        for (int j = i + 1; j < r; j++)
        {
            array[j] = array[j - 1] + 1;
        }

        return true;
    }

    public static bool nextPermutation<T>(T[] array, int start, int length) where T : IComparable<T>
    {
        int end = start + length - 1;

        if (end <= start) return false;

        int last = end;
        while (true)
        {
            int pos = last--;
            
            if (array[last].CompareTo(array[pos]) < 0)
            {
                int i;
                for (i = end + 1; array[last].CompareTo(array[--i]) >= 0; ) { }
                T tmp = array[last]; array[last] = array[i]; array[i] = tmp;
                Array.Reverse(array, pos, end - pos + 1);
                return true;
            }
            if (last == start)
            {
                Array.Reverse(array, start, end - start);
                return false;
            }
        }

        throw new Exception("NextPermutation: Fatal error");
    }

    public static unsafe Dictionary<char, int> cntCharOcc(string s)
    {
        Dictionary<char, int> dict = new Dictionary<char, int>();
        fixed (char* ptr = s)
        {
            for (int i = 0; i < s.Length; i++)
            {
                if (dict.ContainsKey(ptr[i]))
                {
                    dict[ptr[i]] += 1;
                }
                else
                {
                    dict[ptr[i]] = 1;
                }
            }
        }

        return dict;
    }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void chmax<T> (ref T target, T value) where T : struct, IComparisonOperators<T, T, bool>
    {
        if (value > target)
        {
            target = value;
        }
    }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void chmin<T>(ref T target, T value) where T : struct, IComparisonOperators<T, T, bool>
    {
        if (value < target)
        {
            target = value;
        }
    }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void arrMod<T>(T[] array, T b) where T : IModulusOperators<T, T, T>
    {
        for (int i = 0; i < array.Length; i++)
        {
            array[i] %= b;
        }
    }

    public static ReadOnlySpan<T> spanOf<T>(List<T> list)
    {
        return CollectionsMarshal.AsSpan(list);
    }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void swap<T>(T[] array, int i, int j) where T : struct
    {
        (array[i], array[j]) = (array[j], array[i]);
    }

    public static void shuffle<T>(T[] array) where T : struct
    {
        Random random = new Random();
        for (int i = array.Length - 1; i > 0; i--)
        {
            int index = random.Next(0, i + 1);
            swap(array, index, i);
        }
    }

    public static int lowerBound<T>(T[] array, int start, int end, T value) where T : struct, IComparisonOperators<T, T, bool>
    { 
        int low = start;
        int high = end;
        int mid;
        while (low < high)
        {
            mid = ((high - low) >> 1) + low;
            if (array[mid] < value)
                low = mid + 1;
            else
                high = mid;
        }

        if (low == array.Length - 1 && array[low] < value)
        {
            return array.Length;
        }
        return low;
    }

    public static int lowerBound<T>(List<T> array, int start, int end, T value) where T : struct, IComparisonOperators<T, T, bool>
    { 
        int low = start;
        int high = end;
        int mid;
        while (low < high)
        {
            mid = ((high - low) >> 1) + low;
            if (array[mid] < value)
                low = mid + 1;
            else
                high = mid;
        }

        if (low == array.Count - 1 && array[low] < value)
        {
            return array.Count;
        }
        return low;
    }

    public static int upperBound<T>(T[] array, T value) where T : struct, IComparisonOperators<T, T, bool>
    {
        var l = 0;
        var r = array.Length - 1;
        while (l <= r)
        {
            var mid = l + (r - l) / 2;
            if (array[mid] <= value) l = mid + 1;
            else r = mid - 1;
        }
        return l;
    }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void swap<T>(ref T a, ref T b) where T : struct
    {
        T temp = a;
        a = b;
        b = temp;
    }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static bool dbleq(double a, double b)
    {
        return Math.Abs(a - b) < double.Epsilon;
    }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static int dblAsInt(double d)
    {
        return (int)Math.Round(d, 0);
    }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static long dblAsLong(double d)
    {
        return (long)Math.Round(d, 0);
    }

    public static int[] compressCoords<T>(T[] a) where T : struct, IComparisonOperators<T, T, bool>
    {
        T[] b = a.OrderBy(x => x).Distinct().ToArray();
        int[] result = new int[a.Length];

        for (int i = 0; i < a.Length; i++)
        {
            result[i] = lowerBound(b, 0, b.Length - 1, a[i]);
        }

        return result;
    }

    public static List<RunLengthElement<T>> compressRunLength<T>(T[] array) where T : struct, IEquatable<T>
    {
        List<RunLengthElement<T>> list = new List<RunLengthElement<T>>();

        int count = 1;
        int start = 0;
        T prev = array[0];

        for (int i = 1; i < array.Length; i++)
        {
            if (prev.Equals(array[i]))
            {
                count++;
            }
            else
            {
                list.Add(new RunLengthElement<T>(start, count, prev));
                start = i;
                count = 1;
            }

            prev = array[i];
        }

        list.Add(new RunLengthElement<T>(start, count, prev));

        return list;
    }

    public static char[] alphabetsLower()
    {
        return new[] { 'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i', 'j', 'k', 'l', 'm', 'n', 'o', 'p', 'q', 'r', 's', 't', 'u', 'v', 'w', 'x', 'y', 'z' };
    }

    public static char[] alphebetsUpper()
    {
        return new[]{ 'A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M', 'N', 'O', 'P', 'Q', 'R', 'S', 'T', 'U', 'V', 'W', 'X', 'Y', 'Z'};
    }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void print<T>(IEnumerable<T> array, char delimiter = ' ')
    {
        Console.WriteLine(string.Join(delimiter, array));
    }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void fprint<T>(IEnumerable<T> array, char delimiter = ' ')
    {
        print(array);
        Console.Out.Flush();
    }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void print(string msg)
    {
        Console.WriteLine(msg);
    }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void fprint(string msg)
    {
        Console.WriteLine(msg);
        Console.Out.Flush();
    }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void print(int val)
    {
        print(val.ToString());
    }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void print(long val)
    {
        print(val.ToString());
    }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void print(ModInt val)
    {
        print(val.ToString());
    }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void fprint(int val)
    {
        print(val.ToString());
        Console.Out.Flush();
    }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void fprint(long val)
    {
        print(val.ToString());
        Console.Out.Flush();
    }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static void fprint(ModInt val)
    {
        fprint(val.ToString());
        Console.Out.Flush();
    }

    public static void print<T>(T[,] grid)
    {
        for (int y = 0; y < grid.GetLength(0); y++)
        {
            for (int x = 0; x < grid.GetLength(1); x++)
            {
                Console.Write(grid[y, x] + "\t");
            }
            Console.WriteLine();
        }
    }

    public static void print01(bool t)
    {
        Console.WriteLine(t ? "1" : "0");
    }

    public static TSrc lowerBoundKth<TSrc, TCnt>(TSrc min, TSrc max, Func<TSrc, TCnt> f, TCnt bound)
        where TSrc: struct, INumber<TSrc> where TCnt: struct, INumber<TCnt>
    {
        TSrc left = min;
        TSrc right = max;
        TSrc two = TSrc.CreateChecked(2);
        TSrc one = TSrc.CreateChecked(1);
        while (right > left)
        {
            TSrc mid = left + (right - left) / two;
            TCnt c = f(mid);
            if (c < bound)
            {
                left = mid + one;
            }
            else
            {
                right = mid;
            }
        }

        return left;
    }

    public static bool isSquareNumber(long n)
    {
        long q = (long)Math.Floor(Math.Sqrt(n));
        return q * q == n;
    }
}

public readonly struct RunLengthElement<T> where T : struct
{
    public readonly int Count;
    public readonly int StartIndex;
    public readonly T Value;

    public RunLengthElement(int startIndex, int count, T value)
    {
        this.StartIndex = startIndex;
        this.Count = count;
        this.Value = value;
    }
}

public readonly struct Edge<T> : IEquatable<Edge<T>>, IComparable<Edge<T>> where T : struct, INumber<T>
{
    public readonly int To;
    public readonly int From;
    public readonly T Weight;

    public Edge(int to, T weight)
    {
        this.To = to;
        this.Weight = weight;
    }

    public Edge(int from, int to, T weight)
    {
        this.To = to;
        this.From = from;
        this.Weight = weight;
    }

    public override bool Equals(object obj)
    {
        if (obj is Edge<T> edge)
        {
            return this.Equals(edge);
        }
        else
        {
            return false;
        }
    }

    public int CompareTo(Edge<T> other)
    {
        return Weight.CompareTo(other.Weight);
    }

    public bool Equals(Edge<T> edge)
    {
        return To == edge.To && From == edge.From && Weight == edge.Weight;
    }

    public override int GetHashCode()
    {
        return (To, From, Weight).GetHashCode();
    }

    public static bool operator ==(Edge<T> left, Edge<T> right)
    {
        return left.Equals(right);
    }

    public static bool operator !=(Edge<T> left, Edge<T> right)
    {
        return !left.Equals(right);
    }
}

public readonly struct ModInt : IEquatable<ModInt>
{
    private readonly long Value;

    public static ModInt One => (ModInt)1L;

    public static ModInt Zero => (ModInt)0L;

    public ModInt(long value)
    {
        Value = SafeMod(value);
    }

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    private static long SafeMod(long a)
    {
        a %= Constants.Mod;
        if (a < 0) a += Constants.Mod;
        return a;
    }

    public ModInt Power(long exp)
    {
        if (exp <= -1) return this;
        if (exp == 0) return 1;
        if (exp == 1) return this;

        ModInt m = Power(exp / 2);
        m *= m;
        if (exp % 2 == 1) m *= this;

        return m;
    }

    public ModInt Inv()
    {
        return this.Power(Constants.Mod - 2L);
    }

    public static ModInt operator +(ModInt left, ModInt right)
    {
        return new ModInt(SafeMod(left.Value + right.Value));
    }

    public static ModInt operator -(ModInt left, ModInt right)
    {
        return new ModInt(SafeMod(left.Value - right.Value));
    }

    public static ModInt operator *(ModInt left, ModInt right)
    {
        return new ModInt(SafeMod(left.Value * right.Value));
    }

    public static ModInt operator /(ModInt left, ModInt right)
    {
        if (right.Value == 0L)
        {
            return Zero;
        }

        ModInt inv = right.Inv();
        return SafeMod(left * inv);
    }

    public static ModInt operator %(ModInt left, ModInt right)
    {
        if (right.Value == 0L)
        {
            return Zero;
        }

        return new ModInt(SafeMod(left.Value % right.Value));
    }

    public static bool operator ==(ModInt left, ModInt right)
    {
        return left.Value == right.Value;
    }

    public static bool operator != (ModInt left, ModInt right)
    {
        return !(left == right);
    }

    public bool Equals(ModInt other)
    {
        return Value == other.Value;
    }

    public override bool Equals(object other)
    {
        if (other is ModInt m)
        {
            return this == m;
        }
        else return false;
    }

    public override int GetHashCode()
    {
        return Value.GetHashCode();
    }

    public static implicit operator ModInt(long v)
    {
        return new ModInt(v);
    }

    public static implicit operator ModInt(int v)
    {
        return new ModInt(v);
    }

    public static implicit operator long(ModInt m)
    {
        return m.Value;
    }

    public static implicit operator int(ModInt m)
    {
        return (int)m.Value;
    }

    public long Raw() => Value;

    public override string ToString()
    {
        return Value.ToString();
    }
}

public delegate T Monoid<T>(T a, T b);
public delegate T Apply<T, M>(T x, M m, int len);

// 一点に対する操作と区間に対するクエリを処理する.
// 空間計算量: O(2N)
// 時間計算量:
// - 構築: O(N)
// - 操作: O(logN)
// - クエリ: O(logN)
public sealed class SegmentTree<T> where T : struct
{
    private int _treeSize;
    private int _dataSize;
    private int _originalDataSize;
    private T[] _data;
    private Monoid<T> _operator;
    private Monoid<T> _apply;
    private T _identity;

    public int OriginalDataSize => _originalDataSize;
    public int TreeSize => _treeSize;
    public T Identity => _identity;

    public T this[int index]
    {
        get
        {
            return _data[_dataSize - 1 + index];
        }
    }

    public SegmentTree(int n, Monoid<T> op, Monoid<T> apply, T identity)
    {
        _originalDataSize = n;

        int size = 1;
        while (n > size)
        {
            size <<= 1;
        }

        _dataSize = size;
        _treeSize = 2 * size - 1;

        _data = new T[_treeSize];
        _identity = identity;
        _operator = op;
        _apply = apply;
    }

    public void Build(T[] array)
    {
        for (int i = 0; i < array.Length; i++)
        {
            _data[i + _dataSize - 1] = array[i];
        }

        for (int i = _dataSize - 2; i >= 0; i--)
        {
            _data[i] = _operator(_data[(i << 1) + 1], _data[(i << 1) + 2]);
        }
    }

    public void Apply(int index, T value)
    {
        index += _dataSize - 1;
        _data[index] = _apply(_data[index], value);

        while (index > 0)
        {
            index = (index - 1) >> 1;
            _data[index] = _operator(_data[(index << 1) + 1], _data[(index << 1) + 2]);
        }
    }

    public T Query(int left, int right)
    {
        return QueryRec(left, right, 0, 0, _dataSize);
    }

    private T QueryRec(int left, int right, int index, int nodeLeft, int nodeRight)
    {
        if (left >= nodeRight || right <= nodeLeft)
        {
            return _identity;
        }

        if (left <= nodeLeft && nodeRight <= right)
        {
            return _data[index];
        }

        T leftChild = QueryRec(left, right, (index << 1) + 1, nodeLeft, (nodeLeft + nodeRight) >> 1);
        T rightChild = QueryRec(left, right, (index << 1) + 2, (nodeLeft + nodeRight) >> 1, nodeRight);

        return _operator(leftChild, rightChild);
    }

    // 返されたArraySegment<T>は変更してはいけない.
    public ArraySegment<T> GetData()
    {
        return new ArraySegment<T>(_data, _dataSize - 1, _originalDataSize);
    }
}

// 区間に対する操作と区間に対するクエリを処理する.
// 空間計算量: O(4N)
// 時間計算量:
// - 構築: O(N)
// - 操作: O(logN)
// - クエリ: O(logN)
// - 一点参照: O(logN)
// operator: クエリに対応する二項演算をする
// mapping: 作用素を作用させる
// composition: 作用素を合成する
public sealed class LazySegmentTree<T, M> where T : struct, IEquatable<T>  where M : struct, IEquatable<M>
{
    private int _treeSize;
    private int _dataSize;
    private int _originalDataSize;
    private T[] _data;
    private M?[] _lazy;
    private Monoid<T> _operator;
    private Apply<T, M> _mapping;
    private Monoid<M> _composition;
    private T _identity;

    public T this[int index]
    {
        get
        {
            return GetByIndex(index);
        }
    }

    public LazySegmentTree(int n, Monoid<T> op, Apply<T, M> mapping, Monoid<M> composition, T identity)
    {
        _originalDataSize = n;

        int size = 1;
        while (n > size)
        {
            size <<= 1;
        }

        _dataSize = size;
        _treeSize = 2 * size - 1;

        _data = new T[_treeSize];
        _data.AsSpan().Fill(_identity);
        _lazy = new M?[_treeSize];

        _identity = identity;
        _operator = op;
        _mapping = mapping;
        _composition = composition;
    }

    public void Build(T[] array)
    {
        for (int i = 0; i < array.Length; i++)
        {
            _data[i + _dataSize - 1] = array[i];
        }

        for (int i = _dataSize - 2; i >= 0; i--)
        {
            _data[i] = _operator(_data[(i << 1) + 1], _data[(i << 1) + 2]);
        }
    }

    private void Evaluate(int index, int l, int r)
    {
        if (_lazy[index] is null)
        {
            return;
        }
        if (index < _dataSize - 1)
        {
            _lazy[(index << 1) + 1] = GuardComposition(_lazy[(index << 1) + 1], _lazy[index]);
            _lazy[(index << 1) + 2] = GuardComposition(_lazy[(index << 1) + 2], _lazy[index]);
        }

        _data[index] = _mapping(_data[index], (M)_lazy[index], r - l);
        _lazy[index] = null;
    }

    private M GuardComposition(M? a, M? b)
    {
        if (a is null)
        {
            return (M)b;
        }
        else
        {
            return _composition((M)a, (M)b);
        }
    }

    private void ApplyRec(int a, int b, M m, int index, int l, int r)
    {
        Evaluate(index, l, r);

        if (a <= l && r <= b)
        {
            _lazy[index] = GuardComposition(_lazy[index], m);
            Evaluate(index, l, r);
        }
        else if (a < r && l < b)
        {
            ApplyRec(a, b, m, (index << 1) + 1, l, (l + r) / 2);
            ApplyRec(a, b, m, (index << 1) + 2, (l + r) / 2, r);
            _data[index] = _operator(_data[(index << 1) + 1], _data[(index << 1) + 2]);
        }
    }

    public void Apply(int left, int right, M m)
    {
        ApplyRec(left, right, m, 0, 0, _dataSize);
    }

    public T Query(int left, int right)
    {
        return QueryRec(left, right, 0, 0, _dataSize);
    }

    private T QueryRec(int left, int right, int index, int nodeLeft, int nodeRight)
    {
        Evaluate(index, nodeLeft, nodeRight);

        if (left >= nodeRight || right <= nodeLeft)
        {
            return _identity;
        }

        if (left <= nodeLeft && nodeRight <= right)
        {
            return _data[index];
        }

        T leftChild = QueryRec(left, right, (index << 1) + 1, nodeLeft, (nodeLeft + nodeRight) >> 1);
        T rightChild = QueryRec(left, right, (index << 1) + 2, (nodeLeft + nodeRight) >> 1, nodeRight);

        return _operator(leftChild, rightChild);
    }

    public T GetByIndex(int target)
    {
        if (target < 0 || target >= _originalDataSize)
        {
            throw new Exception("Index is out of range.");
        }

        return AccessRec(target, 0, 0, _dataSize);
    }

    private T AccessRec(int target, int index, int l, int r)
    {
        Evaluate(index, l, r);

        if (index >= _dataSize - 1)
        {
            return _data[index];
        }

        int mid = (l + r) / 2;
        if (target < mid)
        {
            return AccessRec(target, (index << 1) + 1, l, mid);
        }
        else
        {
            return AccessRec(target, (index << 1) + 2, mid, r);
        }
    }

    // index以下のノードをすべて即時に評価する.
    private void EvaluateAll(int index, int l, int r)
    {
        if (_lazy[index] is null)
        {
            if (index < _dataSize - 1)
            {
                EvaluateAll((index << 1) + 1, l, (l + r) / 2);
                EvaluateAll((index << 1) + 2, (l + r) / 2, r);
            }
            return;
        }

        if (index < _dataSize - 1)
        {
            _lazy[(index << 1) + 1] = GuardComposition(_lazy[(index << 1) + 1], _lazy[index]);
            _lazy[(index << 1) + 2] = GuardComposition(_lazy[(index << 1) + 2], _lazy[index]);
            EvaluateAll((index << 1) + 1, l, (l + r) / 2);
            EvaluateAll((index << 1) + 2, (l + r) / 2, r);
        }

        _data[index] = _mapping(_data[index], (M)_lazy[index], r - l);
        _lazy[index] = null;
    }

    // 返されたArraySegment<T>は変更してはいけない.
    public ArraySegment<T> GetData()
    {
        EvaluateAll(0, 0, _dataSize);

        return new ArraySegment<T>(_data, _dataSize - 1, _originalDataSize);
    }
}

public struct BeatsNode<T> where T : struct, IEquatable<T>
{
    public T Value;
    public bool Failed;

    public BeatsNode(T value)
    {
        Failed = false;
        Value = value;
    }
}

// 区間に対する操作と区間に対するクエリを処理する.
// 空間計算量: O(4N)
// 時間計算量:
// - 構築: O(N)
// - 操作: (操作に依存した計算量)
// - クエリ: O(logN)
// operator: クエリに対応する二項演算をする. 必ず計算ができる必要がある.
// mapping: 作用素を作用させる. 計算が出来なければFailed←trueとして返す. この際, 子ノードに対する評価が実行される.
// composition: 作用素を合成する
public sealed class SegmentTreeBeats<T, M> where T : struct, IEquatable<T> where M : struct, IEquatable<M>
{
    private int _treeSize;
    private int _dataSize;
    private int _originalDataSize;
    private BeatsNode<T>[] _data;
    private M?[] _lazy;
    private Monoid<BeatsNode<T>> _operator;
    private Apply<BeatsNode<T>, M> _mapping;
    private Monoid<M> _composition;
    private T _identity;
    private BeatsNode<T> _identityNode;

    public T this[int index]
    {
        get{
            return GetByIndex(index);
        }
    }

    public SegmentTreeBeats(int n, Monoid<BeatsNode<T>> op, Apply<BeatsNode<T>, M> mapping, Monoid<M> composition, T identity)
    {
        _originalDataSize = n;

        int size = 1;
        while (n > size)
        {
            size <<= 1;
        }

        _dataSize = size;
        _treeSize = 2 * size - 1;

        _data = new BeatsNode<T>[_treeSize];
        _identityNode = new BeatsNode<T>(_identity);
        _data.AsSpan().Fill(_identityNode);
        _lazy = new M?[_treeSize];

        _identity = identity;
        _operator = op;
        _mapping = mapping;
        _composition = composition;
    }

    public void Build(T[] array)
    {
        for (int i = 0; i < array.Length; i++)
        {
            _data[i + _dataSize - 1] = new BeatsNode<T>(array[i]);
        }

        for (int i = _dataSize - 2; i >= 0; i--)
        {
            _data[i] = _operator(_data[(i << 1) + 1], _data[(i << 1) + 2]);
        }
    }

    private void Evaluate(int index, int l, int r)
    {
        if (_lazy[index] is null)
        {
            return;
        }

        if (index < _dataSize - 1)
        {
            //_lazy[(index << 1) + 1] = GuardComposition(_lazy[(index << 1) + 1], _lazy[index]);
            //_lazy[(index << 1) + 2] = GuardComposition(_lazy[(index << 1) + 2], _lazy[index]);
            if (_lazy[(index << 1) + 1] is not null)
            {
                Evaluate((index << 1) + 1, l, (l + r) / 2);
            }
            _lazy[(index << 1) + 1] = _lazy[index];
            if (_lazy[(index << 1) + 2] is not null)
            {
                Evaluate((index << 1) + 2, (l + r) / 2, r);
            }
            _lazy[(index << 1) + 2] = _lazy[index];
        }

        //_data[index] = _mapping(_data[index], (M)_lazy[index], r - l);

        // 計算してみる
        BeatsNode<T> val = _mapping(_data[index], (M)_lazy[index], r - l);
        
        // 計算出来なければ
        if (val.Failed)
        {
            if (index >= _dataSize - 1)
            {
                throw new Exception("葉ノードに対するmappingは失敗してはいけません");
            }

            // 子ノードを即評価
            Evaluate((index << 1) + 1, l, (l + r) / 2);
            Evaluate((index << 1) + 2, (l + r) / 2, r);

            // 子ノードの結果から親を更新
            _data[index] = _operator(_data[(index << 1) + 1], _data[(index << 1) + 2]);
        }
        else
        {
            // 計算できればそのまま更新
            _data[index] = val;
        }
        _lazy[index] = null;
    }

    private M GuardComposition(M? a, M? b)
    {
        if (a is null)
        {
            return (M)b;
        }
        else
        {
            return _composition((M)a, (M)b);
        }
    }

    private void ApplyRec(int a, int b, M m, int index, int l, int r)
    {
        Evaluate(index, l, r);

        if (a <= l && r <= b)
        {
            _lazy[index] = GuardComposition(_lazy[index], m);
            Evaluate(index, l, r);
        }
        else if (a < r && l < b)
        {
            ApplyRec(a, b, m, (index << 1) + 1, l, (l + r) / 2);
            ApplyRec(a, b, m, (index << 1) + 2, (l + r) / 2, r);
            _data[index] = _operator(_data[(index << 1) + 1], _data[(index << 1) + 2]);
        }
    }

    public void Apply(int left, int right, M m)
    {
        ApplyRec(left, right, m, 0, 0, _dataSize);
    }

    public T Query(int left, int right)
    {
        return QueryRec(left, right, 0, 0, _dataSize).Value;
    }

    private BeatsNode<T> QueryRec(int left, int right, int index, int nodeLeft, int nodeRight)
    {
        Evaluate(index, nodeLeft, nodeRight);

        if (left >= nodeRight || right <= nodeLeft)
        {
            return _identityNode;
        }

        if (left <= nodeLeft && nodeRight <= right)
        {
            return _data[index];
        }

        BeatsNode<T> leftChild = QueryRec(left, right, (index << 1) + 1, nodeLeft, (nodeLeft + nodeRight) >> 1);
        BeatsNode<T> rightChild = QueryRec(left, right, (index << 1) + 2, (nodeLeft + nodeRight) >> 1, nodeRight);

        return _operator(leftChild, rightChild);
    }

    public T GetByIndex(int target)
    {
        if (target < 0 || target >= _originalDataSize)
        {
            throw new Exception("Index is out of range.");
        }

        return AccessRec(target, 0, 0, _dataSize);
    }

    private T AccessRec(int target, int index, int l, int r)
    {
        Evaluate(index, l, r);

        if (index >= _dataSize - 1)
        {
            return _data[index].Value;
        }

        int mid = (l + r) / 2;
        if (target < mid)
        {
            return AccessRec(target, (index << 1) + 1, l, mid);
        }
        else
        {
            return AccessRec(target, (index << 1) + 2, mid, r);
        }
    }

    // index以下のノードをすべて即時に評価する.
    private void EvaluateAll(int index, int l, int r)
    {
        if (_lazy[index] is null)
        {
            if (index < _dataSize - 1)
            {
                EvaluateAll((index << 1) + 1, l, (l + r) / 2);
                EvaluateAll((index << 1) + 2, (l + r) / 2, r);
            }
            return;
        }

        if (index < _dataSize - 1)
        {
            //_lazy[(index << 1) + 1] = GuardComposition(_lazy[(index << 1) + 1], _lazy[index]);
            //_lazy[(index << 1) + 2] = GuardComposition(_lazy[(index << 1) + 2], _lazy[index]);
            if (_lazy[(index << 1) + 1] is not null)
            {
                Evaluate((index << 1) + 1, l, (l + r) / 2);
            }
            _lazy[(index << 1) + 1] = _lazy[index];
            if (_lazy[(index << 1) + 2] is not null)
            {
                Evaluate((index << 1) + 2, (l + r) / 2, r);
            }
            _lazy[(index << 1) + 2] = _lazy[index];
                
            EvaluateAll((index << 1) + 1, l, (l + r) / 2);
            EvaluateAll((index << 1) + 2, (l + r) / 2, r);
        }

        _data[index] = _mapping(_data[index], (M)_lazy[index], r - l);
        _lazy[index] = null;
    }

    // 返されたArraySegment<T>は変更してはいけない.
    public void GetData(T[] destination)
    {
        if (destination.Length < _originalDataSize)
        {
            throw new Exception("書き込み先配列の大きさが足りない");
        }

        EvaluateAll(0, 0, _dataSize);

        //return new ArraySegment<T>(_data, _dataSize - 1, _originalDataSize);
        for (int i = 0; i < _originalDataSize; i++)
        {
            destination[i] = _data[_dataSize - 1 + i].Value;
        }
    }
}

public struct ChangeNode<T> : IEquatable<ChangeNode<T>> where T : struct, INumber<T>, IMinMaxValue<T>
{
    public T Max;
    public int MaxCount;
    public T SecondMax;
    public T Min;
    public int MinCount;
    public T SecondMin;
    public T Sum;

    public bool Equals(ChangeNode<T> other)
    {
        return this.Equals(other);
    }

    public static ChangeNode<R> MakeLeaf<R>(R value) where R : struct, INumber<R>, IMinMaxValue<R>
    {
        return new ChangeNode<R>()
        {
            Max = value,
            MaxCount = 1,
            SecondMax = R.MinValue,
            Min = value,
            MinCount = 1,
            SecondMin = R.MaxValue,
            Sum = value
        };
    }
}

public static class SegmentTreeBuilder
{
    public static LazySegmentTree<T, T> RangeAddMax<T>(int n, T identity) where T : struct, INumber<T>
    {
        return new LazySegmentTree<T, T>(n, T.Max,
        (x, m, l) =>
        {
            return x + m;
        },
        (a, b) => { return a + b; },
        identity);
    }

    public static LazySegmentTree<T, T> RangeUpdateMax<T>(int n, T identity) where T : struct, INumber<T>
    {
        return new LazySegmentTree<T, T>(n, T.Max,
        (x, m, l) =>
        {
            return m;
        },
        (x, y) => { return y; },
        identity);
    }

    public static LazySegmentTree<T, T> RangeAddMin<T>(int n, T identity) where T : struct, INumber<T>
    {
        return new LazySegmentTree<T, T>(n, T.Min,
        (x, m, l) =>
        {
            return x + m;
        },
        (a, b) => { return a + b; },
        identity);
    }

    public static LazySegmentTree<T, T> RangeUpdateMin<T>(int n, T identity) where T : struct, INumber<T>
    {
        return new LazySegmentTree<T, T>(n, T.Min,
        (x, m, l) =>
        {
            return m;
        },
        (x, y) => { return y; },
        identity);
    }

    public static LazySegmentTree<T, T> RangeAddSum<T>(int n, T identity) where T : struct, INumber<T>
    {
        return new LazySegmentTree<T, T>(n, (x, y) => x + y,
        (x, m, l) =>
        {
            return x + m * T.CreateChecked(l);
        },
        (a, b) => { return a + b; },
        identity);
    }

    public static LazySegmentTree<T, T> RangeUpdateSum<T>(int n, T identity) where T : struct, INumber<T>
    {
        return new LazySegmentTree<T, T>(n, (x, y) => x + y,
        (x, m, l) =>
        {
            return m * T.CreateChecked(l);
        },
        (a, b) => { return b; },
        identity);
    }

    public static SegmentTree<T> PointUpdateMax<T>(int n, T identity) where T : struct, INumber<T>
    {
        return new SegmentTree<T>(n, T.Max, (a, b) => b, identity);
    }

    public static SegmentTree<T> PointAddMax<T>(int n, T identity) where T : struct, INumber<T>
    {
        return new SegmentTree<T>(n, T.Max, (a, b) => a + b, identity);
    }

    public static SegmentTree<T> PointUpdateMin<T>(int n, T identity) where T : struct, INumber<T>
    {
        return new SegmentTree<T>(n, T.Min, (a, b) => b, identity);
    }

    public static SegmentTree<T> PointAddMin<T>(int n, T identity) where T : struct, INumber<T>
    {
        return new SegmentTree<T>(n, T.Min, (a, b) => a + b, identity);
    }

    public static SegmentTree<T> PointUpdateSum<T>(int n, T identity) where T : struct, INumber<T>
    {
        return new SegmentTree<T>(n, (x, y) => x + y, (a, b) => b, identity);
    }

    public static SegmentTree<T> PointAddSum<T>(int n, T identity) where T : struct, INumber<T>
    {
        return new SegmentTree<T>(n, (x, y) => x + y, (a, b) => a + b, identity);
    }
}

public sealed class PrefixSum2D<T> where T : struct, IAdditionOperators<T, T, T>, ISubtractionOperators<T, T, T>
{
    private T[,] _sums;

    public PrefixSum2D(T[,] sequence)
    {
        int height = sequence.GetLength(0);
        int width = sequence.GetLength(1);

        _sums = new T[height + 1, width + 1];

        // build prefix sum
        for (int y = 0; y < height; y++)
        {
            for (int x = 0; x < width; x++)
            {
                _sums[y + 1, x + 1] = _sums[y + 1, x] + _sums[y, x + 1] - _sums[y, x] + sequence[y, x];
            }
        }
    }

    public T Sum(int startInclX, int startInclY, int endExclX, int endExclY)
    {
        return _sums[endExclY, endExclX] + _sums[startInclY, startInclX] - _sums[startInclY, endExclX] - _sums[endExclY, startInclX];
    }

    public T AllSum()
    {
        return _sums[_sums.GetLength(0) - 1, _sums.GetLength(1) - 1];
    }
}

public sealed class PrefixSum<T> where T : struct, IAdditionOperators<T, T, T>, ISubtractionOperators<T, T, T>, IAdditiveIdentity<T, T>
{
    private T[] _sums;

    public PrefixSum(T[] sequence)
    {
        _sums = new T[sequence.Length + 1];

        _sums[0] = T.AdditiveIdentity;
        for (int i = 0; i < sequence.Length; i++)
        {
            _sums[i + 1] = _sums[i] + sequence[i];
        }
    }

    // [a, b)
    public T Sum(int aIncl, int bExcl)
    {
        return _sums[bExcl] - _sums[aIncl];
    }

    public T AllSum()
    {
        return _sums[^1];
    }

    public T[] GetArray()
    {
        return _sums;
    }
}

public sealed class UnionFind
{
    private int[] _parents;
    private int[] _size;
    private int _vertexCount;

    public int VertexCount => _vertexCount;

    public UnionFind(int n)
    {
        _vertexCount = n;
        _parents = new int[n];
        _size = new int[n];
        for (int i = 0; i < n; i++)
        {
            _parents[i] = i;
            _size[i] = 1;
        }
    }

    public int Root(int x)
    {
        if (_parents[x] == x)
            return x;
        return _parents[x] = Root(_parents[x]);
    }

    public void Unite(int x, int y)
    {
        int rootX = Root(x);
        int rootY = Root(y);
        if (rootX == rootY)
            return;

        int from = rootX;
        int to = rootY;
        // merge from to to
        if (_size[from] > _size[to])
        {
            (from, to) = (to, from);
        }

        _size[to] += _size[from];
        _parents[from] = to;
    }

    public List<int> Find(int x)
    {
        int rootX = Root(x);
        List<int> set = new List<int>();
        for (int i = 0; i < _vertexCount; i++)
        {
            if (Root(i) == rootX)
                set.Add(i);
        }

        return set;
    }

    public Dictionary<int, List<int>> FindAll()
    {
        Dictionary<int, List<int>> sets = new Dictionary<int, List<int>>();
        for (int i = 0; i < _vertexCount; i++)
        {
            int root = Root(i);
            if (sets.ContainsKey(root))
                sets[root].Add(i);
            else
                sets[root] = new List<int>() { i };
        }

        return sets;
    }

    public bool Same(int x, int y)
    {
        int rootX = Root(x);
        int rootY = Root(y);
        return rootX == rootY;
    }

    public int Size(int v)
    {
        return _size[Root(v)];
    }

    public void Clear()
    {
        for (int i = 0; i < _vertexCount; i++)
        {
            _parents[i] = i;
        }
    }
}

public sealed class RollbackUnionFind
{
    private int[] _data;
    private Stack<(int, int)> _history;
    int _innerSnap;

    public RollbackUnionFind(int n)
    {
        _innerSnap = 0;
        _data = new int[n];
        Array.Fill(_data, -1);
        _history = new Stack<(int, int)>();
    }

    public bool Unite(int x, int y)
    {
        x = Find(x);
        y = Find(y);

        _history.Push((x, _data[x]));
        _history.Push((y, _data[y]));

        if (x == y)
            return false;
        
        if (_data[x] > _data[y])
        {
            swap(ref x, ref y);
        }

        _data[x] += _data[y];
        _data[y] = x;
        return true;
    }

    public int Find(int k)
    {
        if (_data[k] < 0) return k;
        return Find(_data[k]);
    }

    public bool Same(int x, int y)
    {
        return Find(x) == Find(y);
    }

    public int Size(int k)
    {
        return -_data[Find(k)];
    }

    public void Undo()
    {
        _data[_history.Peek().Item1] = _history.Peek().Item2;
        _history.Pop();
        _data[_history.Peek().Item1] = _history.Peek().Item2;
        _history.Pop();
    }

    public void Snapshot()
    {
        _innerSnap = _history.Count >> 1;
    }

    public int GetState()
    {
        return _history.Count >> 1;
    }

    public void Rollback(int state = -1)
    {
        if (state == -1) state = _innerSnap;
        state <<= 1;
        while (state < _history.Count) this.Undo();
    }
}

public sealed class Imos<T> where T : struct, IAdditionOperators<T, T, T>, ISubtractionOperators<T, T, T>
{
    private T[] _data;
    

    public Imos(T[] array)
    {
        _data = array;
    }

    public Imos(int len)
    {
        _data = new T[len];
    }

    public void AddQueryLen(int start, int length, T value)
    {
        this.AddQuery(start, start + length, value);
    }

    public void AddQuery(int start, int end, T value)
    {
        _data[start] += value;
        if (end < _data.Length)
        {
            _data[end] -= value;
        }
    }

    public void Simulate()
    {
        for (int i = 1; i < _data.Length; i++)
        {
            _data[i] += _data[i - 1];
        }
    }

    public T[] GetData()
    {
        return _data;
    }
}

public sealed class Imos2D<T> where T : struct, IAdditionOperators<T, T, T>, ISubtractionOperators<T, T, T>
{
    private T[,] _data;
    private int _width;
    private int _height;

    public Imos2D(T[,] data)
    {
        _data = data;
        _width = data.GetLength(1);
        _height = data.GetLength(0);
    }

    public Imos2D(int h, int w)
    {
        _data = new T[h, w];
        _width = w;
        _height = h;
    }

    public void AddQuery(int startX, int startY, int endX, int endY, T value)
    {
        _data[startY, startX] += value;
        if (endX < _width)
        {
            _data[startY, endX] -= value;
        }
        if (endY < _height)
        {
            _data[endY, startX] -= value;
        }
        if (endX < _width && endY < _height)
        {
            _data[endY, endX] += value;
        }
    }

    public void AddQueryLen(int x, int y, int w, int h, T value)
    {
        this.AddQuery(x, y, x + w, y + h, value);
    }

    public void Simulate()
    {
        for (int x = 1; x < _width; x++)
        {
            for (int y = 0; y < _height; y++)
            {
                _data[y, x] += _data[y, x - 1];
            }
        }

        for (int y = 1; y < _height; y++)
        {
            for (int x = 0; x < _width; x++)
            {
                _data[y, x] += _data[y - 1, x];
            }
        }
    }

    public T[,] GetData()
    {
        return _data;
    }
}

public static class LIS
{
    // NOTE: 空間計算量O(max), 時間計算量O(nlogn)
    public static int LengthOfLIS(int[] sequence, int max)
    {
        if (max >= 1_000_000_00)
        {
            throw new ArgumentException("Max value is too large.", nameof(max));
        }

        var seg = SegmentTreeBuilder.PointUpdateMax<int>(max + 1, 0);

        for (int i = 1; i <= sequence.Length; i++)
        {
            int m = seg.Query(0, sequence[i - 1]);
            seg.Apply(sequence[i - 1], m + 1);
        }

        return seg.Query(0, max + 1);
    }
}

public static class TestCase
{
    private static Random _random = new Random();

    public static void Seed(int seed)
    {
        _random = new Random(seed);
    }
    
    public static string RandomStringLower(int length)
    {
        Span<char> span = stackalloc char[length];
        for (int i = 0; i < length; i++)
        {
            char ch = (char)('a' + _random.Next(0, 26));
            span[i] = ch;
        }
        return new string(span);
    }

    public static string RandomStringUpper(int length)
    {
        return RandomStringLower(length).ToUpperInvariant();
    }

    public static int[] RandomPermutation(int length)
    {
        int[] d = new int[length];
        for (int i = 0; i < length; i++) d[i] = i + 1;
        shuffle(d);
        return d;
    }

    public static long[] RandomArrayLong(int length, long min, long max)
    {
        long[] d = new long[length];
        for (int i = 0; i < length; i++)
        {
            d[i] = _random.NextInt64(min, max + 1);
        }

        return d;
    }

    public static int[] RandomArray(int length, int min, int max)
    {
        int[] d = new int[length];
        for (int i = 0; i < length; i++)
        {
            d[i] = _random.Next(min, max + 1);
        }

        return d;
    }

    public static int[,] Random2DArray(int height, int width, int min, int max)
    {
        int[,] d = new int[height, width];
        for (int y = 0; y < height; y++)
        {
            for (int x = 0; x < width; x++)
            {
                d[y, x] = _random.Next(min, max + 1);
            }
        }

        return d;
    }

    public static long[,] Random2DArrayLong(int height, int width, int min, int max)
    {
        long[,] d = new long[height, width];
        for (int y = 0; y < height; y++)
        {
            for (int x = 0; x < width; x++)
            {
                d[y, x] = _random.NextInt64(min, max + 1);
            }
        }

        return d;
    }

    public static char[,] RandomCharGrid(int height, int width, params char[] tiles)
    {
        char[,] d = new char[height, width];

        for (int y = 0; y < height; y++)
        {
            for (int x = 0; x < width; x++)
            {
                d[y, x] = tiles[_random.Next(0, tiles.Length)];
            }
        }

        return d;
    }
}

public sealed class NumberUtility
{
    private bool[] _isPrime;
    private int[] _minFactor;
    private int[] _mobius;
    private int _n;

    public NumberUtility(int max)
    {
        _n = max;

        _isPrime = new bool[max + 1];
        _minFactor = new int[max + 1];
        _mobius = new int[max + 1];

        Array.Fill(_isPrime, true);
        Array.Fill(_minFactor, -1);
        Array.Fill(_mobius, 1);

        _isPrime[1] = false;
        _minFactor[1] = 1;

        for (int i = 2; i <= _n; i++)
        {
            if (!_isPrime[i]) continue;

            _minFactor[i] = i;
            _mobius[i] = -1;

            for (int j = i + i; j <= _n; j += i)
            {
                _isPrime[j] = false;

                if (_minFactor[j] == -1) _minFactor[j] = i;
                if (j / i % i == 0) _mobius[j] = 0;
                else _mobius[j] = -_mobius[j];
            }
        }
    }

    public int Mu(int n)
    {
        return _mobius[n];
    }

    public int MinFactor(int n)
    {
        return _minFactor[n];
    }

    public bool IsPrime(int n)
    {
        return _isPrime[n];
    }
}

public sealed class Convolution
{
    private readonly long _mod;

    public Convolution(long mod)
    {
        _mod = mod;
    }

    public List<long> CalcConvolution(List<long> a, List<long> b)
    {
        return null;
    }
}