using System;
using System.Diagnostics;
using System.Numerics;
using System.Runtime.CompilerServices;
using System.Runtime.Intrinsics.X86;
#region いつもの
using AtCoderProject;
using System;
using System.Collections.Generic;
using System.Linq;
using static AtCoderProject.Global;
using BigInteger = System.Numerics.BigInteger;
using BitOperations = System.Numerics.BitOperations;
using IEnumerable = System.Collections.IEnumerable;
using IEnumerator = System.Collections.IEnumerator;
using StringBuilder = System.Text.StringBuilder;
using Unsafe = System.Runtime.CompilerServices.Unsafe;
using AtCoder;
using Math = AtCoder.Math;

namespace AtCoderProject
{
    using System.IO;
    using System.Text;
    using System.Diagnostics;
    using System.Runtime.CompilerServices;
    using System.Linq.Expressions;

    public static class Global
    {
        public static int Gcd(params int[] nums)
        {
            var gcd = nums[0];
            for (var i = 1; i < nums.Length; i++)
                gcd = Gcd(nums[i], gcd);
            return gcd;
        }
        public static int Gcd(int a, int b) => b > a ? Gcd(b, a) : (b == 0 ? a : Gcd(b, a % b));
        public static int Lcm(int a, int b) => a / Gcd(a, b) * b;
        public static int Lcm(params int[] nums)
        {
            var lcm = nums[0];
            for (var i = 1; i < nums.Length; i++)
                lcm = Lcm(lcm, nums[i]);
            return lcm;
        }
        public static long Gcd(params long[] nums)
        {
            var gcd = nums[0];
            for (var i = 1; i < nums.Length; i++)
                gcd = Gcd(nums[i], gcd);
            return gcd;
        }
        public static long Gcd(long a, long b) => b > a ? Gcd(b, a) : (b == 0 ? a : Gcd(b, a % b));
        public static long Lcm(long a, long b) => a / Gcd(a, b) * b;

        public static long Lcm(params long[] nums)
        {
            var lcm = nums[0];
            for (var i = 1; i < nums.Length; i++)
                lcm = Lcm(lcm, nums[i]);
            return lcm;
        }

        public static T[] NewArray<T>(int len0, T value) => new T[len0].Fill(value);
        public static T[] NewArray<T>(int len0, Func<T> factory)
        {
            var arr = new T[len0];
            for (int i = 0; i < arr.Length; i++) arr[i] = factory();
            return arr;
        }
        public static T[][] NewArray<T>(int len0, int len1, T value) where T : struct
        {
            var arr = new T[len0][];
            for (int i = 0; i < arr.Length; i++) arr[i] = NewArray(len1, value);
            return arr;
        }
        public static T[][] NewArray<T>(int len0, int len1, Func<T> factory)
        {
            var arr = new T[len0][];
            for (int i = 0; i < arr.Length; i++) arr[i] = NewArray(len1, factory);
            return arr;
        }
        public static T[][][] NewArray<T>(int len0, int len1, int len2, T value) where T : struct
        {
            var arr = new T[len0][][];
            for (int i = 0; i < arr.Length; i++) arr[i] = NewArray(len1, len2, value);
            return arr;
        }
        public static T[][][] NewArray<T>(int len0, int len1, int len2, Func<T> factory)
        {
            var arr = new T[len0][][];
            for (int i = 0; i < arr.Length; i++) arr[i] = NewArray(len1, len2, factory);
            return arr;
        }
        public static T[][][][] NewArray<T>(int len0, int len1, int len2, int len3, T value) where T : struct
        {
            var arr = new T[len0][][][];
            for (int i = 0; i < arr.Length; i++) arr[i] = NewArray(len1, len2, len3, value);
            return arr;
        }
        public static T[][][][] NewArray<T>(int len0, int len1, int len2, int len3, Func<T> factory)
        {
            var arr = new T[len0][][][];
            for (int i = 0; i < arr.Length; i++) arr[i] = NewArray(len1, len2, len3, factory);
            return arr;
        }


        public static long Pow(long x, int y)
        {
            long res = 1;
            for (; y > 0; y >>= 1)
            {
                if ((y & 1) == 1) res *= x;
                x *= x;
            }
            return res;
        }
        public static BigInteger ParseBigInteger(ReadOnlySpan<char> s)
        {
            /* 自前実装の方が速い */
            if (s[0] == '-') return -ParseBigInteger(s[1..]);
            BigInteger res;
            if (s.Length % 9 == 0)
                res = 0;
            else
            {
                res = new BigInteger(int.Parse(s.Slice(0, s.Length % 9)));
                s = s.Slice(s.Length % 9);
            }

            while (s.Length > 0)
            {
                var sp = s.Slice(0, 9);
                res *= 1000_000_000;
                res += int.Parse(sp);
                s = s.Slice(9);
            }
            return res;
        }
        public static int PopCount(int x) => BitOperations.PopCount((uint)x);
        public static int PopCount(long x) => BitOperations.PopCount((ulong)x);
        public static int PopCount(ulong x) => BitOperations.PopCount(x);
        public static int MSB(int x) => BitOperations.Log2((uint)x);
        public static int MSB(uint x) => BitOperations.Log2(x);
        public static int MSB(long x) => BitOperations.Log2((ulong)x);
        public static int MSB(ulong x) => BitOperations.Log2(x);
        public static int LSB(int x) => BitOperations.TrailingZeroCount((uint)x);
        public static int LSB(uint x) => BitOperations.TrailingZeroCount(x);
        public static int LSB(long x) => BitOperations.TrailingZeroCount((ulong)x);
        public static int LSB(ulong x) => BitOperations.TrailingZeroCount(x);

    }
    public static class Ext
    {

        public static bool UpdateMax<T>(this ref T r, T val) where T : struct, IComparable<T>
        {
            if (r.CompareTo(val) < 0) { r = val; return true; }
            return false;
        }
        public static bool UpdateMin<T>(this ref T r, T val) where T : struct, IComparable<T>
        {
            if (r.CompareTo(val) > 0) { r = val; return true; }
            return false;
        }
        public static T[] Fill<T>(this T[] arr, T value)
        {
            Array.Fill(arr, value);
            return arr;
        }
        public static T[] Sort<T>(this T[] arr) { Array.Sort(arr); return arr; }
        public static string[] Sort(this string[] arr) => Sort(arr, StringComparer.Ordinal);
        public static T[] Sort<T, U>(this T[] arr, Expression<Func<T, U>> selector) where U : IComparable<U> => Sort(arr, ExComparer<T>.CreateExp(selector));
        public static T[] Sort<T>(this T[] arr, Comparison<T> comparison) { Array.Sort(arr, comparison); return arr; }
        public static T[] Sort<T>(this T[] arr, IComparer<T> comparer) { Array.Sort(arr, comparer); return arr; }
        public static T[] Reverse<T>(this T[] arr) { Array.Reverse(arr); return arr; }
        public static (int index, T max) MaxBy<T>(this T[] arr) where T : IComparable<T>
        {
            T max = arr[0];
            int maxIndex = 0;
            for (int i = 0; i < arr.Length; i++)
            {
                if (max.CompareTo(arr[i]) < 0)
                {
                    max = arr[i];
                    maxIndex = i;
                }
            }
            return (maxIndex, max);
        }
        public static (int index, T max) MaxBy<T, TMax>(this T[] arr, Func<T, TMax> maxBySelector) where TMax : IComparable<TMax>
        {
            var maxItem = maxBySelector(arr[0]);
            var max = arr[0];
            int maxIndex = 0;
            for (int i = 0; i < arr.Length; i++)
            {
                var nx = maxBySelector(arr[i]);
                if (maxItem.CompareTo(nx) < 0)
                {
                    maxItem = nx;
                    max = arr[i];
                    maxIndex = i;
                }
            }
            return (maxIndex, max);
        }
        public static (TSource item, TMax max) MaxBy<TSource, TMax>
            (this IEnumerable<TSource> source, Func<TSource, TMax> maxBySelector)
            where TMax : IComparable<TMax>
        {
            TMax max;
            TSource maxByItem;
            var e = source.GetEnumerator();
            e.MoveNext();
            maxByItem = e.Current;
            max = maxBySelector(maxByItem);
            while (e.MoveNext())
            {
                var item = e.Current;
                var next = maxBySelector(item);
                if (max.CompareTo(next) < 0)
                {
                    max = next;
                    maxByItem = item;
                }
            }
            return (maxByItem, max);
        }
        public static (int index, T min) MinBy<T>(this T[] arr) where T : IComparable<T>
        {
            T min = arr[0];
            int minIndex = 0;
            for (int i = 0; i < arr.Length; i++)
            {
                if (min.CompareTo(arr[i]) > 0)
                {
                    min = arr[i];
                    minIndex = i;
                }
            }
            return (minIndex, min);
        }
        public static (int index, T min) MinBy<T, TMin>(this T[] arr, Func<T, TMin> minBySelector) where TMin : IComparable<TMin>
        {
            var minItem = minBySelector(arr[0]);
            var min = arr[0];
            int minIndex = 0;
            for (int i = 0; i < arr.Length; i++)
            {
                var nx = minBySelector(arr[i]);
                if (minItem.CompareTo(nx) > 0)
                {
                    minItem = nx;
                    min = arr[i];
                    minIndex = i;
                }
            }
            return (minIndex, min);
        }
        public static (TSource item, TMin min) MinBy<TSource, TMin>
            (this IEnumerable<TSource> source, Func<TSource, TMin> minBySelector)
            where TMin : IComparable<TMin>
        {
            TMin min;
            TSource minByItem;

            var e = source.GetEnumerator();
            e.MoveNext();
            minByItem = e.Current;
            min = minBySelector(minByItem);
            while (e.MoveNext())
            {
                var item = e.Current;
                var next = minBySelector(item);
                if (min.CompareTo(next) > 0)
                {
                    min = next;
                    minByItem = item;
                }
            }
            return (minByItem, min);
        }
        public static Dictionary<TKey, int> GroupCount<TSource, TKey>(this IEnumerable<TSource> source, Func<TSource, TKey> keySelector) => source.GroupBy(keySelector).ToDictionary(g => g.Key, g => g.Count());
        public static Dictionary<TKey, int> GroupCount<TKey>(this IEnumerable<TKey> source) => source.GroupCount(i => i);
        public static Span<T> AsSpan<T>(this List<T> list, int start = 0) => Unsafe.As<Tuple<T[]>>(list).Item1.AsSpan(start, list.Count);
        public static ref T Get<T>(this T[] arr, int index)
        {
            if (index < 0)
                return ref arr[arr.Length + index];
            return ref arr[index];
        }
        public static TValue Get<TKey, TValue>(this IDictionary<TKey, TValue> dic, TKey key)
        {
            dic.TryGetValue(key, out var v);
            return v;
        }
    }
    public static class ExComparer<T>
    {
        class ExpressionComparer<K> : IComparer<T> where K : IComparable<K>
        {
            private class ParameterReplaceVisitor : ExpressionVisitor
            {
                private readonly ParameterExpression from;
                private readonly ParameterExpression to;
                public ParameterReplaceVisitor(ParameterExpression from, ParameterExpression to)
                {
                    this.from = from;
                    this.to = to;
                }
                protected override Expression VisitParameter(ParameterExpression node) => node == from ? to : base.VisitParameter(node);
            }

            readonly Comparison<T> func;
            public ExpressionComparer(Expression<Func<T, K>> expression)
            {
                var paramA = expression.Parameters[0];
                var paramB = Expression.Parameter(typeof(T));
                var f2 = (Expression<Func<T, K>>)new ParameterReplaceVisitor(expression.Parameters[0], paramB).Visit(expression);
                var compExp = Expression.Lambda<Comparison<T>>(Expression.Call(
                        expression.Body,
                        typeof(K).GetMethod(nameof(IComparable<K>.CompareTo), new[] { typeof(K) }),
                        f2.Body),
                        paramA, paramB);
                this.func = compExp.Compile();
            }
            public int Compare(T x, T y) => func(x, y);
            public override bool Equals(object obj) => obj is ExpressionComparer<K> c && this.func == c.func;
            public override int GetHashCode() => func.GetHashCode();
        }
        class ReverseComparer : IComparer<T>
        {
            private static Comparer<T> orig = Comparer<T>.Default;
            public int Compare(T y, T x) => orig.Compare(x, y);
            public override bool Equals(object obj) => obj is ReverseComparer;
            public override int GetHashCode() => GetType().GetHashCode();
        }
        public static IComparer<T> CreateExp<K>(Expression<Func<T, K>> expression) where K : IComparable<K> => new ExpressionComparer<K>(expression);
        public static readonly IComparer<T> DefaultReverse = new ReverseComparer();
    }
    public class ΔDebugView<T> { private IEnumerable<T> collection; public ΔDebugView(IEnumerable<T> collection) { this.collection = collection ?? throw new ArgumentNullException(nameof(collection)); }[DebuggerBrowsable(DebuggerBrowsableState.RootHidden)] public T[] Items => collection.ToArray(); }


    [DebuggerStepThrough]
    public class ConsoleReader
    {
        const int BufSize = 1 << 12;
        private readonly byte[] buffer = new byte[BufSize];
        private readonly Stream input;
        private readonly Encoding encoding;
        private int pos = 0;
        private int len = 0;
        public ConsoleReader(Stream input, Encoding encoding) { this.input = input; this.encoding = encoding; }
        public ConsoleReader() : this(Console.OpenStandardInput(), Console.InputEncoding) { }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        private void MoveNext() { if (++pos >= len) { len = input.Read(buffer, 0, buffer.Length); if (len == 0) { buffer[0] = 10; } pos = 0; } }

        public int Int
        {
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            get
            {
                int res = 0;
                bool neg = false;
                while (buffer[pos] < 48) { neg = buffer[pos] == 45; MoveNext(); }
                do { res = checked(res * 10 + (buffer[pos] ^ 48)); MoveNext(); } while (48 <= buffer[pos]);
                return neg ? -res : res;
            }
        }
        public int Int0 => this.Int - 1;
        public long Long
        {
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            get
            {
                long res = 0;
                bool neg = false;
                while (buffer[pos] < 48) { neg = buffer[pos] == 45; MoveNext(); }
                do { res = res * 10 + (buffer[pos] ^ 48); MoveNext(); } while (48 <= buffer[pos]);
                return neg ? -res : res;
            }
        }
        public long Long0 => this.Long - 1;
        public string String
        {
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            get
            {
                var sb = new List<byte>();
                while (buffer[pos] <= 32) MoveNext();
                do { sb.Add(buffer[pos]); MoveNext(); } while (32 < buffer[pos]);
                return this.encoding.GetString(sb.ToArray());
            }
        }
        public string Ascii
        {
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            get
            {
                var sb = new StringBuilder();
                while (buffer[pos] <= 32) MoveNext();
                do { sb.Append((char)buffer[pos]); MoveNext(); } while (32 < buffer[pos]);
                return sb.ToString();
            }
        }
        public string Line
        {
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            get
            {
                var sb = new List<byte>();
                while (buffer[pos] <= 32) MoveNext();
                do { sb.Add(buffer[pos]); MoveNext(); } while (buffer[pos] != 10 && buffer[pos] != 13);
                return this.encoding.GetString(sb.ToArray());
            }
        }
        public char Char
        {
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            get
            {
                while (buffer[pos] <= 32) MoveNext();
                char res = (char)buffer[pos];
                MoveNext();
                return res;
            }
        }
        public double Double => double.Parse(this.Ascii);

        [DebuggerStepThrough]
        public struct RepeatReader
        {
            ConsoleReader cr;
            int count;
            public RepeatReader(ConsoleReader cr, int count) { this.cr = cr; this.count = count; }
            public T[] Select<T>(Func<ConsoleReader, T> factory) { var arr = new T[count]; for (var i = 0; i < count; i++) arr[i] = factory(cr); return arr; }
            public T[] Select<T>(Func<ConsoleReader, int, T> factory) { var arr = new T[count]; for (var i = 0; i < count; i++) arr[i] = factory(cr, i); return arr; }
            public string[] Line { get { var arr = new string[count]; for (var i = 0; i < count; i++) arr[i] = cr.Line; return arr; } }
            public string[] String { get { var arr = new string[count]; for (var i = 0; i < count; i++) arr[i] = cr.String; return arr; } }
            public string[] Ascii { get { var arr = new string[count]; for (var i = 0; i < count; i++) arr[i] = cr.Ascii; return arr; } }
            public int[] Int { get { var arr = new int[count]; for (var i = 0; i < count; i++) arr[i] = cr.Int; return arr; } }
            public int[] Int0 { get { var arr = new int[count]; for (var i = 0; i < count; i++) arr[i] = cr.Int0; return arr; } }
            public long[] Long { get { var arr = new long[count]; for (var i = 0; i < count; i++) arr[i] = cr.Long; return arr; } }
            public long[] Long0 { get { var arr = new long[count]; for (var i = 0; i < count; i++) arr[i] = cr.Long0; return arr; } }
            public double[] Double { get { var arr = new double[count]; for (var i = 0; i < count; i++) arr[i] = cr.Double; return arr; } }

            public static implicit operator int[](RepeatReader rr) => rr.Int;
            public static implicit operator long[](RepeatReader rr) => rr.Long;
            public static implicit operator double[](RepeatReader rr) => rr.Double;
            public static implicit operator string[](RepeatReader rr) => rr.Ascii;
        }
        public RepeatReader Repeat(int count) => new RepeatReader(this, count);

        [DebuggerStepThrough]
        public struct SplitReader
        {
            ConsoleReader cr;
            public SplitReader(ConsoleReader cr) { this.cr = cr; }
            public string[] String { get { while (cr.buffer[cr.pos] <= 32) cr.MoveNext(); var list = new List<string>(); do { if (cr.buffer[cr.pos] < 32) cr.MoveNext(); else list.Add(cr.String); } while (cr.buffer[cr.pos] != 10 && cr.buffer[cr.pos] != 13); return list.ToArray(); } }
            public string[] Ascii { get { while (cr.buffer[cr.pos] <= 32) cr.MoveNext(); var list = new List<string>(); do { if (cr.buffer[cr.pos] < 32) cr.MoveNext(); else list.Add(cr.Ascii); } while (cr.buffer[cr.pos] != 10 && cr.buffer[cr.pos] != 13); return list.ToArray(); } }
            public int[] Int { get { while (cr.buffer[cr.pos] <= 32) cr.MoveNext(); var list = new List<int>(); do { if (cr.buffer[cr.pos] < 32) cr.MoveNext(); else list.Add(cr.Int); } while (cr.buffer[cr.pos] != 10 && cr.buffer[cr.pos] != 13); return list.ToArray(); } }
            public int[] Int0 { get { while (cr.buffer[cr.pos] <= 32) cr.MoveNext(); var list = new List<int>(); do { if (cr.buffer[cr.pos] < 32) cr.MoveNext(); else list.Add(cr.Int0); } while (cr.buffer[cr.pos] != 10 && cr.buffer[cr.pos] != 13); return list.ToArray(); } }
            public long[] Long { get { while (cr.buffer[cr.pos] <= 32) cr.MoveNext(); var list = new List<long>(); do { if (cr.buffer[cr.pos] < 32) cr.MoveNext(); else list.Add(cr.Long); } while (cr.buffer[cr.pos] != 10 && cr.buffer[cr.pos] != 13); return list.ToArray(); } }
            public long[] Long0 { get { while (cr.buffer[cr.pos] <= 32) cr.MoveNext(); var list = new List<long>(); do { if (cr.buffer[cr.pos] < 32) cr.MoveNext(); else list.Add(cr.Long0); } while (cr.buffer[cr.pos] != 10 && cr.buffer[cr.pos] != 13); return list.ToArray(); } }
            public double[] Double { get { while (cr.buffer[cr.pos] <= 32) cr.MoveNext(); var list = new List<double>(); do { if (cr.buffer[cr.pos] < 32) cr.MoveNext(); else list.Add(cr.Double); } while (cr.buffer[cr.pos] != 10 && cr.buffer[cr.pos] != 13); return list.ToArray(); } }

            public static implicit operator int[](SplitReader sr) => sr.Int;
            public static implicit operator long[](SplitReader sr) => sr.Long;
            public static implicit operator double[](SplitReader sr) => sr.Double;
            public static implicit operator string[](SplitReader sr) => sr.Ascii;
        }
        public SplitReader Split => new SplitReader(this);

        public static implicit operator int(ConsoleReader cr) => cr.Int;
        public static implicit operator long(ConsoleReader cr) => cr.Long;
        public static implicit operator double(ConsoleReader cr) => cr.Double;
        public static implicit operator string(ConsoleReader cr) => cr.Ascii;
    }
    [DebuggerStepThrough]
    public class ConsoleWriter
    {
        [DebuggerBrowsable(DebuggerBrowsableState.Never)] public readonly StreamWriter sw;
        public ConsoleWriter() : this(Console.OpenStandardOutput(), Console.OutputEncoding) { }
        public ConsoleWriter(Stream output, Encoding encoding) { sw = new StreamWriter(output, encoding); }
        public void Flush() => sw.Flush();
        public ConsoleWriter WriteLine(ReadOnlySpan<char> obj) { sw.WriteLine(obj); return this; }
        public ConsoleWriter WriteLine<T>(T obj) { sw.WriteLine(obj.ToString()); return this; }
        public ConsoleWriter WriteLineJoin<T>(Span<T> col) => WriteMany(' ', (ReadOnlySpan<T>)col);
        public ConsoleWriter WriteLineJoin<T>(ReadOnlySpan<T> col) => WriteMany(' ', col);
        public ConsoleWriter WriteLineJoin<T>(IEnumerable<T> col) => WriteMany(' ', col);
        public ConsoleWriter WriteLineJoin<T>(params T[] col) => WriteMany(' ', col);
        public ConsoleWriter WriteLineJoin(params object[] col) => WriteMany(' ', col);
        public ConsoleWriter WriteLineJoin<T1, T2>(T1 v1, T2 v2)
        {
            sw.Write(v1.ToString()); sw.Write(' ');
            sw.WriteLine(v2.ToString()); return this;
        }
        public ConsoleWriter WriteLineJoin<T1, T2, T3>(T1 v1, T2 v2, T3 v3)
        {
            sw.Write(v1.ToString()); sw.Write(' ');
            sw.Write(v2.ToString()); sw.Write(' ');
            sw.WriteLine(v3.ToString()); return this;
        }
        public ConsoleWriter WriteLineJoin<T1, T2, T3, T4>(T1 v1, T2 v2, T3 v3, T4 v4)
        {
            sw.Write(v1.ToString()); sw.Write(' ');
            sw.Write(v2.ToString()); sw.Write(' ');
            sw.Write(v3.ToString()); sw.Write(' ');
            sw.WriteLine(v4.ToString()); return this;
        }
        public ConsoleWriter WriteLines<T>(Span<T> col) => WriteMany('\n', (ReadOnlySpan<T>)col);
        public ConsoleWriter WriteLines<T>(ReadOnlySpan<T> col) => WriteMany('\n', col);
        public ConsoleWriter WriteLines<T>(IEnumerable<T> col) => WriteMany('\n', col);
        public ConsoleWriter WriteLineGrid<T>(IEnumerable<IEnumerable<T>> cols)
        {
            foreach (var col in cols)
                WriteLineJoin(col);
            return this;
        }
        private ConsoleWriter WriteMany<T>(char sep, ReadOnlySpan<T> col)
        {
            var en = col.GetEnumerator();
            if (!en.MoveNext())
                return this;
            sw.Write(en.Current.ToString());
            while (en.MoveNext())
            {
                sw.Write(sep);
                sw.Write(en.Current.ToString());
            }
            sw.WriteLine();
            return this;
        }
        private ConsoleWriter WriteMany<T>(char sep, IEnumerable<T> col)
        {
            var en = col.GetEnumerator();
            if (!en.MoveNext())
                return this;
            sw.Write(en.Current.ToString());
            while (en.MoveNext())
            {
                sw.Write(sep);
                sw.Write(en.Current.ToString());
            }
            sw.WriteLine();
            return this;
        }
    }
}
public partial class Program
{
    [System.Diagnostics.DebuggerBrowsable(System.Diagnostics.DebuggerBrowsableState.Never)] public ConsoleReader cr;
    [System.Diagnostics.DebuggerBrowsable(System.Diagnostics.DebuggerBrowsableState.Never)] public ConsoleWriter cw;
    public Program(ConsoleReader reader, ConsoleWriter writer) { this.cr = reader; this.cw = writer; System.Globalization.CultureInfo.CurrentCulture = System.Globalization.CultureInfo.InvariantCulture; }
    static void Main() => new Program(new ConsoleReader(), new ConsoleWriter()).Run();
    public void Run()
    {
        var res = Calc();
        if (res is double)
            cw.WriteLine(Result((double)res));
        else if (res is bool)
            cw.WriteLine(Result((bool)res));
        else if (res != null)
            cw.WriteLine(res.ToString());
        cw.Flush();
    }
}
public partial class Program
{
    public static string Result(double d) => d.ToString("0.####################", System.Globalization.CultureInfo.InvariantCulture);
    #endregion
    public static string Result(bool b) => b ? "Yes" : "No";
    private object Calc()
    {
        int n = cr;
        int m = cr;

        var a = new StaticModInt<Mod998244353>[n];
        var b = new StaticModInt<Mod998244353>[m];

        for (int i = 0; i < n; i++)
        {
            a[i] = cr.Int;
        }
        for (int i = 0; i < m; i++)
        {
            b[i] = cr.Int;
        }

        var c = Math.Convolution(a, b);
        cw.WriteLines(c);
        return null;
    }
}

namespace AtCoder
{
    public static partial class Math
    {
        /// <summary>
        /// 畳み込みを mod <paramref name="m"/> = 998244353 で計算します。
        /// </summary>
        /// <remarks>
        /// <para><paramref name="a"/>, <paramref name="b"/> の少なくとも一方が空の場合は空配列を返します。</para>
        /// <para>制約:</para>
        /// <para>- 2≤<paramref name="m"/>≤2×10^9</para>
        /// <para>- <paramref name="m"/> は素数</para>
        /// <para>- 2^c | (<paramref name="m"/> - 1) かつ |<paramref name="a"/>| + |<paramref name="b"/>| - 1 ≤ 2^c なる c が存在する</para>
        /// <para>計算量: O((|<paramref name="a"/>|+|<paramref name="b"/>|)log(|<paramref name="a"/>|+|<paramref name="b"/>|) + log<paramref name="m"/>)</para>
        /// </remarks>
        public static int[] Convolution(int[] a, int[] b) => Convolution<Mod998244353>(a, b);

        /// <summary>
        /// 畳み込みを mod <paramref name="m"/> = 998244353 で計算します。
        /// </summary>
        /// <remarks>
        /// <para><paramref name="a"/>, <paramref name="b"/> の少なくとも一方が空の場合は空配列を返します。</para>
        /// <para>制約:</para>
        /// <para>- 2≤<paramref name="m"/>≤2×10^9</para>
        /// <para>- <paramref name="m"/> は素数</para>
        /// <para>- 2^c | (<paramref name="m"/> - 1) かつ |<paramref name="a"/>| + |<paramref name="b"/>| - 1 ≤ 2^c なる c が存在する</para>
        /// <para>計算量: O((|<paramref name="a"/>|+|<paramref name="b"/>|)log(|<paramref name="a"/>|+|<paramref name="b"/>|) + log<paramref name="m"/>)</para>
        /// </remarks>
        public static uint[] Convolution(uint[] a, uint[] b) => Convolution<Mod998244353>(a, b);

        /// <summary>
        /// 畳み込みを mod <paramref name="m"/> = 998244353 で計算します。
        /// </summary>
        /// <remarks>
        /// <para><paramref name="a"/>, <paramref name="b"/> の少なくとも一方が空の場合は空配列を返します。</para>
        /// <para>制約:</para>
        /// <para>- 2≤<paramref name="m"/>≤2×10^9</para>
        /// <para>- <paramref name="m"/> は素数</para>
        /// <para>- 2^c | (<paramref name="m"/> - 1) かつ |<paramref name="a"/>| + |<paramref name="b"/>| - 1 ≤ 2^c なる c が存在する</para>
        /// <para>計算量: O((|<paramref name="a"/>|+|<paramref name="b"/>|)log(|<paramref name="a"/>|+|<paramref name="b"/>|) + log<paramref name="m"/>)</para>
        /// </remarks>
        public static long[] Convolution(long[] a, long[] b) => Convolution<Mod998244353>(a, b);

        /// <summary>
        /// 畳み込みを mod <paramref name="m"/> = 998244353 で計算します。
        /// </summary>
        /// <remarks>
        /// <para><paramref name="a"/>, <paramref name="b"/> の少なくとも一方が空の場合は空配列を返します。</para>
        /// <para>制約:</para>
        /// <para>- 2≤<paramref name="m"/>≤2×10^9</para>
        /// <para>- <paramref name="m"/> は素数</para>
        /// <para>- 2^c | (<paramref name="m"/> - 1) かつ |<paramref name="a"/>| + |<paramref name="b"/>| - 1 ≤ 2^c なる c が存在する</para>
        /// <para>計算量: O((|<paramref name="a"/>|+|<paramref name="b"/>|)log(|<paramref name="a"/>|+|<paramref name="b"/>|) + log<paramref name="m"/>)</para>
        /// </remarks>
        public static ulong[] Convolution(ulong[] a, ulong[] b) => Convolution<Mod998244353>(a, b);

        /// <summary>
        /// 畳み込みを mod <typeparamref name="TMod"/> で計算します。
        /// </summary>
        /// <remarks>
        /// <para><paramref name="a"/>, <paramref name="b"/> の少なくとも一方が空の場合は空配列を返します。</para>
        /// <para>制約:</para>
        /// <para>- 2≤<typeparamref name="TMod"/>≤2×10^9</para>
        /// <para>- <typeparamref name="TMod"/> は素数</para>
        /// <para>- 2^c | (<typeparamref name="TMod"/> - 1) かつ |<paramref name="a"/>| + |<paramref name="b"/>| - 1 ≤ 2^c なる c が存在する</para>
        /// <para>計算量: O((|<paramref name="a"/>|+|<paramref name="b"/>|)log(|<paramref name="a"/>|+|<paramref name="b"/>|) + log<typeparamref name="TMod"/>)</para>
        /// </remarks>
        public static int[] Convolution<TMod>(int[] a, int[] b) where TMod : struct, IStaticMod
        {
            var n = a.Length;
            var m = b.Length;
            if (n == 0 || m == 0)
            {
                return Array.Empty<int>();
            }

            if (System.Math.Min(n, m) <= 60)
            {
                var c = ConvolutionNaive<TMod>(a.Select(ai => new StaticModInt<TMod>(ai)).ToArray(),
                                               b.Select(bi => new StaticModInt<TMod>(bi)).ToArray());
                return c.Select(ci => ci.Value).ToArray();
            }
            else
            {
                int z = 1 << Internal.InternalMath.CeilPow2(n + m - 1);

                var aTemp = new StaticModInt<TMod>[z];
                for (int i = 0; i < a.Length; i++)
                {
                    aTemp[i] = new StaticModInt<TMod>(a[i]);
                }

                var bTemp = new StaticModInt<TMod>[z];
                for (int i = 0; i < b.Length; i++)
                {
                    bTemp[i] = new StaticModInt<TMod>(b[i]);
                }

                var c = Convolution<TMod>(aTemp, bTemp, n, m, z)[0..(n + m - 1)];
                var result = new int[c.Length];
                for (int i = 0; i < result.Length; i++)
                {
                    result[i] = c[i].Value;
                }
                return result;
            }
        }


        /// <summary>
        /// 畳み込みを mod <typeparamref name="TMod"/> で計算します。
        /// </summary>
        /// <remarks>
        /// <para><paramref name="a"/>, <paramref name="b"/> の少なくとも一方が空の場合は空配列を返します。</para>
        /// <para>制約:</para>
        /// <para>- 2≤<typeparamref name="TMod"/>≤2×10^9</para>
        /// <para>- <typeparamref name="TMod"/> は素数</para>
        /// <para>- 2^c | (<typeparamref name="TMod"/> - 1) かつ |<paramref name="a"/>| + |<paramref name="b"/>| - 1 ≤ 2^c なる c が存在する</para>
        /// <para>計算量: O((|<paramref name="a"/>|+|<paramref name="b"/>|)log(|<paramref name="a"/>|+|<paramref name="b"/>|) + log<typeparamref name="TMod"/>)</para>
        /// </remarks>
        public static uint[] Convolution<TMod>(uint[] a, uint[] b) where TMod : struct, IStaticMod
        {
            var n = a.Length;
            var m = b.Length;
            if (n == 0 || m == 0)
            {
                return Array.Empty<uint>();
            }

            if (System.Math.Min(n, m) <= 60)
            {
                var c = ConvolutionNaive<TMod>(a.Select(ai => new StaticModInt<TMod>(ai)).ToArray(),
                                               b.Select(bi => new StaticModInt<TMod>(bi)).ToArray());
                return c.Select(ci => (uint)ci.Value).ToArray();
            }
            else
            {
                int z = 1 << Internal.InternalMath.CeilPow2(n + m - 1);

                var aTemp = new StaticModInt<TMod>[z];
                for (int i = 0; i < a.Length; i++)
                {
                    aTemp[i] = new StaticModInt<TMod>(a[i]);
                }

                var bTemp = new StaticModInt<TMod>[z];
                for (int i = 0; i < b.Length; i++)
                {
                    bTemp[i] = new StaticModInt<TMod>(b[i]);
                }

                var c = Convolution<TMod>(aTemp, bTemp, n, m, z)[0..(n + m - 1)];
                var result = new uint[c.Length];
                for (int i = 0; i < result.Length; i++)
                {
                    result[i] = (uint)c[i].Value;
                }
                return result;
            }
        }

        /// <summary>
        /// 畳み込みを mod <typeparamref name="TMod"/> で計算します。
        /// </summary>
        /// <remarks>
        /// <para><paramref name="a"/>, <paramref name="b"/> の少なくとも一方が空の場合は空配列を返します。</para>
        /// <para>制約:</para>
        /// <para>- 2≤<typeparamref name="TMod"/>≤2×10^9</para>
        /// <para>- <typeparamref name="TMod"/> は素数</para>
        /// <para>- 2^c | (<typeparamref name="TMod"/> - 1) かつ |<paramref name="a"/>| + |<paramref name="b"/>| - 1 ≤ 2^c なる c が存在する</para>
        /// <para>計算量: O((|<paramref name="a"/>|+|<paramref name="b"/>|)log(|<paramref name="a"/>|+|<paramref name="b"/>|) + log<typeparamref name="TMod"/>)</para>
        /// </remarks>
        public static long[] Convolution<TMod>(long[] a, long[] b) where TMod : struct, IStaticMod
        {
            var n = a.Length;
            var m = b.Length;
            if (n == 0 || m == 0)
            {
                return Array.Empty<long>();
            }

            if (System.Math.Min(n, m) <= 60)
            {
                var c = ConvolutionNaive<TMod>(a.Select(ai => new StaticModInt<TMod>(ai)).ToArray(),
                                               b.Select(bi => new StaticModInt<TMod>(bi)).ToArray());
                return c.Select(ci => (long)ci.Value).ToArray();
            }
            else
            {
                int z = 1 << Internal.InternalMath.CeilPow2(n + m - 1);

                var aTemp = new StaticModInt<TMod>[z];
                for (int i = 0; i < a.Length; i++)
                {
                    aTemp[i] = new StaticModInt<TMod>(a[i]);
                }

                var bTemp = new StaticModInt<TMod>[z];
                for (int i = 0; i < b.Length; i++)
                {
                    bTemp[i] = new StaticModInt<TMod>(b[i]);
                }

                var c = Convolution<TMod>(aTemp, bTemp, n, m, z)[0..(n + m - 1)];
                var result = new long[c.Length];
                for (int i = 0; i < result.Length; i++)
                {
                    result[i] = c[i].Value;
                }
                return result;
            }
        }

        /// <summary>
        /// 畳み込みを mod <typeparamref name="TMod"/> で計算します。
        /// </summary>
        /// <remarks>
        /// <para><paramref name="a"/>, <paramref name="b"/> の少なくとも一方が空の場合は空配列を返します。</para>
        /// <para>制約:</para>
        /// <para>- 2≤<typeparamref name="TMod"/>≤2×10^9</para>
        /// <para>- <typeparamref name="TMod"/> は素数</para>
        /// <para>- 2^c | (<typeparamref name="TMod"/> - 1) かつ |<paramref name="a"/>| + |<paramref name="b"/>| - 1 ≤ 2^c なる c が存在する</para>
        /// <para>計算量: O((|<paramref name="a"/>|+|<paramref name="b"/>|)log(|<paramref name="a"/>|+|<paramref name="b"/>|) + log<typeparamref name="TMod"/>)</para>
        /// </remarks>
        public static ulong[] Convolution<TMod>(ulong[] a, ulong[] b) where TMod : struct, IStaticMod
        {
            var n = a.Length;
            var m = b.Length;
            if (n == 0 || m == 0)
            {
                return Array.Empty<ulong>();
            }

            if (System.Math.Min(n, m) <= 60)
            {
                var c = ConvolutionNaive<TMod>(a.Select(TakeMod).ToArray(),
                                               b.Select(TakeMod).ToArray());
                return c.Select(ci => (ulong)ci.Value).ToArray();
            }
            else
            {
                int z = 1 << Internal.InternalMath.CeilPow2(n + m - 1);

                var aTemp = new StaticModInt<TMod>[z];
                for (int i = 0; i < a.Length; i++)
                {
                    aTemp[i] = TakeMod(a[i]);
                }

                var bTemp = new StaticModInt<TMod>[z];
                for (int i = 0; i < b.Length; i++)
                {
                    bTemp[i] = TakeMod(b[i]);
                }

                var c = Convolution<TMod>(aTemp, bTemp, n, m, z)[0..(n + m - 1)];
                var result = new ulong[c.Length];
                for (int i = 0; i < result.Length; i++)
                {
                    result[i] = (ulong)c[i].Value;
                }
                return result;
            }

            StaticModInt<TMod> TakeMod(ulong x) => StaticModInt<TMod>.Raw((int)(x % default(TMod).Mod));
        }

        /// <summary>
        /// 畳み込みを mod <typeparamref name="TMod"/> で計算します。
        /// </summary>
        /// <remarks>
        /// <para><paramref name="a"/>, <paramref name="b"/> の少なくとも一方が空の場合は空配列を返します。</para>
        /// <para>制約:</para>
        /// <para>- 2≤<typeparamref name="TMod"/>≤2×10^9</para>
        /// <para>- <typeparamref name="TMod"/> は素数</para>
        /// <para>- 2^c | (<typeparamref name="TMod"/> - 1) かつ |<paramref name="a"/>| + |<paramref name="b"/>| - 1 ≤ 2^c なる c が存在する</para>
        /// <para>計算量: O((|<paramref name="a"/>|+|<paramref name="b"/>|)log(|<paramref name="a"/>|+|<paramref name="b"/>|) + log<typeparamref name="TMod"/>)</para>
        /// </remarks>
        public static StaticModInt<TMod>[] Convolution<TMod>(StaticModInt<TMod>[] a, StaticModInt<TMod>[] b)
            where TMod : struct, IStaticMod
        {
            var temp = Convolution((ReadOnlySpan<StaticModInt<TMod>>)a, b);
            return temp.ToArray();
        }

        /// <summary>
        /// 畳み込みを mod <typeparamref name="TMod"/> で計算します。
        /// </summary>
        /// <remarks>
        /// <para><paramref name="a"/>, <paramref name="b"/> の少なくとも一方が空の場合は空配列を返します。</para>
        /// <para>制約:</para>
        /// <para>- 2≤<typeparamref name="TMod"/>≤2×10^9</para>
        /// <para>- <typeparamref name="TMod"/> は素数</para>
        /// <para>- 2^c | (<typeparamref name="TMod"/> - 1) かつ |<paramref name="a"/>| + |<paramref name="b"/>| - 1 ≤ 2^c なる c が存在する</para>
        /// <para>計算量: O((|<paramref name="a"/>|+|<paramref name="b"/>|)log(|<paramref name="a"/>|+|<paramref name="b"/>|) + log<typeparamref name="TMod"/>)</para>
        /// </remarks>
        public static Span<StaticModInt<TMod>> Convolution<TMod>(ReadOnlySpan<StaticModInt<TMod>> a, ReadOnlySpan<StaticModInt<TMod>> b)
            where TMod : struct, IStaticMod
        {
            var n = a.Length;
            var m = b.Length;
            if (n == 0 || m == 0)
            {
                return Array.Empty<StaticModInt<TMod>>();
            }

            if (System.Math.Min(n, m) <= 60)
            {
                return ConvolutionNaive(a, b);
            }

            int z = 1 << Internal.InternalMath.CeilPow2(n + m - 1);

            var aTemp = new StaticModInt<TMod>[z];
            a.CopyTo(aTemp);

            var bTemp = new StaticModInt<TMod>[z];
            b.CopyTo(bTemp);

            return Convolution(aTemp.AsSpan(), bTemp.AsSpan(), n, m, z);
        }

        private static Span<StaticModInt<TMod>> Convolution<TMod>(Span<StaticModInt<TMod>> a, Span<StaticModInt<TMod>> b, int n, int m, int z)
            where TMod : struct, IStaticMod
        {
            Internal.Butterfly<TMod>.Calculate(a);
            Internal.Butterfly<TMod>.Calculate(b);

            for (int i = 0; i < a.Length; i++)
            {
                a[i] *= b[i];
            }

            Internal.Butterfly<TMod>.CalculateInv(a);
            var result = a[0..(n + m - 1)];
            var iz = new StaticModInt<TMod>(z).Inv();
            foreach (ref var r in result)
            {
                r *= iz;
            }

            return result;
        }

        /// <summary>
        /// 畳み込みを計算します。
        /// </summary>
        /// <remarks>
        /// <para><paramref name="a"/>, <paramref name="b"/> の少なくとも一方が空の場合は空配列を返します。</para>
        /// <para>制約:</para>
        /// <para>- |<paramref name="a"/>| + |<paramref name="b"/>| - 1 ≤ 2^24 = 16,777,216</para>
        /// <para>- 畳み込んだ後の配列の要素が全て long に収まる</para>
        /// <para>計算量: O((|<paramref name="a"/>|+|<paramref name="b"/>|)log(|<paramref name="a"/>|+|<paramref name="b"/>|))</para>
        /// </remarks>
        public static long[] ConvolutionLong(ReadOnlySpan<long> a, ReadOnlySpan<long> b)
        {
            unchecked
            {
                var n = a.Length;
                var m = b.Length;

                if (n == 0 || m == 0)
                {
                    return Array.Empty<long>();
                }

                const ulong Mod1 = 754974721;
                const ulong Mod2 = 167772161;
                const ulong Mod3 = 469762049;
                const ulong M2M3 = Mod2 * Mod3;
                const ulong M1M3 = Mod1 * Mod3;
                const ulong M1M2 = Mod1 * Mod2;
                // (m1 * m2 * m3) % 2^64
                const ulong M1M2M3 = Mod1 * Mod2 * Mod3;

                ulong i1 = (ulong)Internal.InternalMath.InvGCD((long)M2M3, (long)Mod1).Item2;
                ulong i2 = (ulong)Internal.InternalMath.InvGCD((long)M1M3, (long)Mod2).Item2;
                ulong i3 = (ulong)Internal.InternalMath.InvGCD((long)M1M2, (long)Mod3).Item2;

                var c1 = Convolution<FFTMod1>(a, b);
                var c2 = Convolution<FFTMod2>(a, b);
                var c3 = Convolution<FFTMod3>(a, b);

                var c = new long[n + m - 1];

                Span<ulong> offset = stackalloc ulong[] { 0, 0, M1M2M3, 2 * M1M2M3, 3 * M1M2M3 };

                for (int i = 0; i < c.Length; i++)
                {
                    ulong x = 0;
                    x += (c1[i] * i1) % Mod1 * M2M3;
                    x += (c2[i] * i2) % Mod2 * M1M3;
                    x += (c3[i] * i3) % Mod3 * M1M2;

                    long diff = (long)c1[i] - Internal.InternalMath.SafeMod((long)x, (long)Mod1);
                    if (diff < 0)
                    {
                        diff += (long)Mod1;
                    }

                    // 真値を r, 得られた値を x, M1M2M3 % 2^64 = M', B = 2^63 として、
                    // r = x,
                    //     x -  M' + (0 or 2B),
                    //     x - 2M' + (0 or 2B or 4B),
                    //     x - 3M' + (0 or 2B or 4B or 6B)
                    // のいずれかが成り立つ、らしい
                    // -> see atcoder/convolution.hpp
                    x -= offset[(int)(diff % offset.Length)];
                    c[i] = (long)x;
                }

                return c;
            }


            ulong[] Convolution<TMod>(ReadOnlySpan<long> a, ReadOnlySpan<long> b) where TMod : struct, IStaticMod
            {
                int z = 1 << Internal.InternalMath.CeilPow2(a.Length + b.Length - 1);

                var aTemp = new StaticModInt<TMod>[z];
                for (int i = 0; i < a.Length; i++)
                {
                    aTemp[i] = new StaticModInt<TMod>(a[i]);
                }

                var bTemp = new StaticModInt<TMod>[z];
                for (int i = 0; i < b.Length; i++)
                {
                    bTemp[i] = new StaticModInt<TMod>(b[i]);
                }

                var c = AtCoder.Math.Convolution<TMod>(aTemp, bTemp, a.Length, b.Length, z);
                var result = new ulong[c.Length];
                for (int i = 0; i < result.Length; i++)
                {
                    result[i] = (ulong)c[i].Value;
                }

                return result;
            }
        }

        private static StaticModInt<TMod>[] ConvolutionNaive<TMod>(ReadOnlySpan<StaticModInt<TMod>> a, ReadOnlySpan<StaticModInt<TMod>> b)
            where TMod : struct, IStaticMod
        {
            if (a.Length < b.Length)
            {
                // ref 構造体のため型引数として使えない
                var temp = a;
                a = b;
                b = temp;
            }

            var ans = new StaticModInt<TMod>[a.Length + b.Length - 1];
            for (int i = 0; i < a.Length; i++)
            {
                for (int j = 0; j < b.Length; j++)
                {
                    ans[i + j] += a[i] * b[j];
                }
            }

            return ans;
        }

        private readonly struct FFTMod1 : IStaticMod
        {
            public uint Mod => 754974721;
            public bool IsPrime => true;
        }

        private readonly struct FFTMod2 : IStaticMod
        {
            public uint Mod => 167772161;
            public bool IsPrime => true;
        }

        private readonly struct FFTMod3 : IStaticMod
        {
            public uint Mod => 469762049;
            public bool IsPrime => true;
        }
    }
}
namespace AtCoder.Internal
{
    public static class Butterfly<T> where T : struct, IStaticMod
    {
        /// <summary>
        /// sumE[i] = ies[0] * ... * ies[i - 1] * es[i]
        /// </summary>
        private static StaticModInt<T>[] sumE;

        /// <summary>
        /// sumIE[i] = es[0] * ... * es[i - 1] * ies[i]
        /// </summary>
        private static StaticModInt<T>[] sumIE;

        [MethodImpl(MethodImplOptions.AggressiveOptimization)]
        public static void Calculate(Span<StaticModInt<T>> a)
        {
            var n = a.Length;
            var h = InternalMath.CeilPow2(n);

            // 一度計算したらキャッシュしておく
            sumE ??= CalcurateSumE();

            for (int ph = 1; ph <= h; ph++)
            {
                // ブロックサイズの半分
                int w = 1 << (ph - 1);

                // ブロック数
                int p = 1 << (h - ph);

                var now = StaticModInt<T>.Raw(1);

                // 各ブロックの s 段目
                for (int s = 0; s < w; s++)
                {
                    int offset = s << (h - ph + 1);

                    var ls = a.Slice(offset, p);
                    var rs = a.Slice(offset + p, p);

                    for (int i = 0; i < p; i++)
                    {
                        var l = ls[i];
                        var r = rs[i] * now;
                        ls[i] = l + r;
                        rs[i] = l - r;
                    }
                    now *= sumE[InternalMath.BSF(~(uint)s)];
                }
            }

            static StaticModInt<T>[] CalcurateSumE()
            {
                int g = InternalMath.PrimitiveRoot((int)default(T).Mod);
                int cnt2 = InternalMath.BSF(default(T).Mod - 1);
                var e = new StaticModInt<T>(g).Pow((default(T).Mod - 1) >> cnt2);
                var ie = e.Inv();

                var sumE = new StaticModInt<T>[cnt2 - 2];

                // es[i]^(2^(2+i)) == 1
                Span<StaticModInt<T>> es = stackalloc StaticModInt<T>[cnt2 - 1];
                Span<StaticModInt<T>> ies = stackalloc StaticModInt<T>[cnt2 - 1];

                for (int i = es.Length - 1; i >= 0; i--)
                {
                    // e^(2^(2+i)) == 1
                    es[i] = e;
                    ies[i] = ie;
                    e *= e;
                    ie *= ie;
                }

                var now = StaticModInt<T>.Raw(1);
                for (int i = 0; i < sumE.Length; i++)
                {
                    sumE[i] = es[i] * now;
                    now *= ies[i];
                }

                return sumE;
            }
        }

        [MethodImpl(MethodImplOptions.AggressiveOptimization)]
        public static void CalculateInv(Span<StaticModInt<T>> a)
        {
            var n = a.Length;
            var h = InternalMath.CeilPow2(n);

            // 一度計算したらキャッシュしておく
            sumIE ??= CalcurateSumIE();

            for (int ph = h; ph >= 1; ph--)
            {
                // ブロックサイズの半分
                int w = 1 << (ph - 1);

                // ブロック数
                int p = 1 << (h - ph);

                var iNow = StaticModInt<T>.Raw(1);

                // 各ブロックの s 段目
                for (int s = 0; s < w; s++)
                {
                    int offset = s << (h - ph + 1);

                    var ls = a.Slice(offset, p);
                    var rs = a.Slice(offset + p, p);

                    for (int i = 0; i < p; i++)
                    {
                        var l = ls[i];
                        var r = rs[i];
                        ls[i] = l + r;
                        rs[i] = (l - r) * iNow;
                    }
                    iNow *= sumIE[InternalMath.BSF(~(uint)s)];
                }
            }

            static StaticModInt<T>[] CalcurateSumIE()
            {
                int g = InternalMath.PrimitiveRoot((int)default(T).Mod);
                int cnt2 = InternalMath.BSF(default(T).Mod - 1);
                var e = new StaticModInt<T>(g).Pow((default(T).Mod - 1) >> cnt2);
                var ie = e.Inv();

                var sumIE = new StaticModInt<T>[cnt2 - 2];

                // es[i]^(2^(2+i)) == 1
                Span<StaticModInt<T>> es = stackalloc StaticModInt<T>[cnt2 - 1];
                Span<StaticModInt<T>> ies = stackalloc StaticModInt<T>[cnt2 - 1];

                for (int i = es.Length - 1; i >= 0; i--)
                {
                    // e^(2^(2+i)) == 1
                    es[i] = e;
                    ies[i] = ie;
                    e *= e;
                    ie *= ie;
                }

                var now = StaticModInt<T>.Raw(1);
                for (int i = 0; i < sumIE.Length; i++)
                {
                    sumIE[i] = ies[i] * now;
                    now *= es[i];
                }

                return sumIE;
            }
        }
    }
}

namespace AtCoder.Internal
{
    public static partial class InternalMath
    {
        /// <summary>
        /// <paramref name="n"/> ≤ 2**x を満たす最小のx
        /// </summary>
        /// <remarks>
        /// <para>制約: 0≤<paramref name="n"/></para>
        /// </remarks>
        public static int CeilPow2(int n)
        {
            var un = (uint)n;
            if (un <= 1) return 0;
            return BitOperations.Log2(un - 1) + 1;
        }
    }
}

namespace AtCoder.Internal
{
    public static partial class InternalMath
    {
        /// <summary>
        /// (<paramref name="n"/> &amp; (1 &lt;&lt; x)) != 0 なる最小の非負整数 x を求めます。
        /// </summary>
        /// <remarks>
        /// <para>BSF: Bit Scan Forward</para>
        /// <para>制約: 1 ≤ <paramref name="n"/></para>
        /// </remarks>
        public static int BSF(uint n)
        {
            Debug.Assert(n >= 1);
            if (Bmi1.IsSupported)
            {
                // O(1)
                return (int)Bmi1.TrailingZeroCount(n);
            }
            else if (Popcnt.IsSupported)
            {
                // O(1)
                return (int)Popcnt.PopCount(~n & (n - 1));
            }
            else
            {
                // O(logn)
                return BitOperations.TrailingZeroCount(n);
            }
        }
    }
}

namespace AtCoder.Internal
{
    public static partial class InternalMath
    {
        public static long PowMod(long x, long n, int m)
        {
            Debug.Assert(n >= 0);
            Debug.Assert(m >= 1);

            if (m == 1)
            {
                return 0;
            }

            uint _m = (uint)m;
            ulong r = 1;
            ulong y = (ulong)SafeMod(x, m);

            while (n > 0)
            {
                if ((n & 1) > 0)
                {
                    r = (r * y) % _m;
                }

                y = (y * y) % _m;
                n >>= 1;
            }

            return (long)r;
        }
    }
}

namespace AtCoder.Internal
{
    /// <summary>
    /// Fast moduler by barrett reduction
    /// <seealso href="https://en.wikipedia.org/wiki/Barrett_reduction"/>
    /// </summary>
    public class Barrett
    {
        public uint Mod { get; private set; }
        private ulong IM;
        public Barrett(uint m)
        {
            Mod = m;
            IM = unchecked((ulong)-1) / m + 1;
        }

        /// <summary>
        /// <paramref name="a"/> * <paramref name="b"/> mod m
        /// </summary>
        public uint Mul(uint a, uint b)
        {
            ulong z = a;
            z *= b;
            if (!Bmi2.X64.IsSupported) return (uint)(z % Mod);
            var x = Bmi2.X64.MultiplyNoFlags(z, IM);
            var v = unchecked((uint)(z - x * Mod));
            if (Mod <= v) v += Mod;
            return v;
        }
    }
}

namespace AtCoder.Internal
{
    public static partial class InternalMath
    {
        public static int PrimitiveRoot(int m)
        {
            Debug.Assert(m >= 2);

            return m switch
            {
                2 => 1,
                167772161 => 3,
                469762049 => 3,
                754974721 => 11,
                998244353 => 3,
                _ => Calculate(m)
            };

            int Calculate(int m)
            {
                Span<int> divs = stackalloc int[20];
                divs[0] = 2;
                int cnt = 1;
                int x = (m - 1) / 2;

                while (x % 2 == 0)
                {
                    x >>= 1;
                }

                for (int i = 3; (long)i * i <= x; i += 2)
                {
                    if (x % i == 0)
                    {
                        divs[cnt++] = i;
                        while (x % i == 0)
                        {
                            x /= i;
                        }
                    }
                }

                if (x > 1)
                {
                    divs[cnt++] = x;
                }

                for (int g = 2; ; g++)
                {
                    bool ok = true;
                    for (int i = 0; i < cnt; i++)
                    {
                        if (PowMod(g, (m - 1) / divs[i], m) == 1)
                        {
                            ok = false;
                            break;
                        }
                    }

                    if (ok)
                    {
                        return g;
                    }
                }
            }
        }
    }
}

namespace AtCoder.Internal
{
    public static partial class InternalMath
    {
        public static long SafeMod(long x, long m)
        {
            x %= m;
            if (x < 0) x += m;
            return x;
        }
    }
}

namespace AtCoder.Internal
{
    public static partial class InternalMath
    {
        /// <summary>
        /// g=gcd(a,b),xa=g(mod b) となるような 0≤x&lt;b/g の(g, x)
        /// </summary>
        /// <remarks>
        /// <para>制約: 1≤<paramref name="b"/></para>
        /// </remarks>
        public static (long, long) InvGCD(long a, long b)
        {
            a = SafeMod(a, b);
            if (a == 0) return (b, 0);

            long s = b, t = a;
            long m0 = 0, m1 = 1;

            long u;
            while (true)
            {
                if (t == 0)
                {
                    if (m0 < 0) m0 += b / s;
                    return (s, m0);
                }
                u = s / t;
                s -= t * u;
                m0 -= m1 * u;

                if (s == 0)
                {
                    if (m1 < 0) m1 += b / t;
                    return (t, m1);
                }
                u = t / s;
                t -= s * u;
                m1 -= m0 * u;
            }
        }
    }
}

namespace AtCoder
{
    /// <summary>
    /// コンパイル時に決定する mod を表します。
    /// </summary>
    /// <example>
    /// <code>
    /// public readonly struct Mod1000000009 : IStaticMod
    /// {
    ///     public uint Mod => 1000000009;
    ///     public bool IsPrime => true;
    /// }
    /// </code>
    /// </example>
    public interface IStaticMod
    {
        /// <summary>
        /// mod を取得します。
        /// </summary>
        uint Mod { get; }

        /// <summary>
        /// mod が素数であるか識別します。
        /// </summary>
        bool IsPrime { get; }
    }

    public readonly struct Mod1000000007 : IStaticMod
    {
        public uint Mod => 1000000007;
        public bool IsPrime => true;
    }

    public readonly struct Mod998244353 : IStaticMod
    {
        public uint Mod => 998244353;
        public bool IsPrime => true;
    }

    /// <summary>
    /// 実行時に決定する mod の ID を表します。
    /// </summary>
    /// <example>
    /// <code>
    /// public readonly struct ModID123 : IDynamicModID { }
    /// </code>
    /// </example>
    public interface IDynamicModID { }

    public readonly struct ModID0 : IDynamicModID { }
    public readonly struct ModID1 : IDynamicModID { }
    public readonly struct ModID2 : IDynamicModID { }

    /// <summary>
    /// 四則演算時に自動で mod を取る整数型。mod の値はコンパイル時に決定している必要があります。
    /// </summary>
    /// <typeparam name="T">定数 mod を表す構造体</typeparam>
    /// <example>
    /// <code>
    /// using ModInt = AtCoder.StaticModInt&lt;AtCoder.Mod1000000007&gt;;
    ///
    /// void SomeMethod()
    /// {
    ///     var m = new ModInt(1);
    ///     m -= 2;
    ///     Console.WriteLine(m);   // 1000000006
    /// }
    /// </code>
    /// </example>
    public readonly struct StaticModInt<T> where T : struct, IStaticMod
    {
        private readonly uint _v;

        /// <summary>
        /// 格納されている値を返します。
        /// </summary>
        public int Value => (int)_v;

        /// <summary>
        /// mod を返します。
        /// </summary>
        public static int Mod => (int)default(T).Mod;

        /// <summary>
        /// <paramref name="v"/> に対して mod を取らずに StaticModInt&lt;<typeparamref name="T"/>&gt; 型のインスタンスを生成します。
        /// </summary>
        /// <remarks>
        /// <para>定数倍高速化のための関数です。 <paramref name="v"/> に 0 未満または mod 以上の値を入れた場合の挙動は未定義です。</para>
        /// <para>制約: 0≤|<paramref name="v"/>|&lt;mod</para>
        /// </remarks>
        [MethodImpl(MethodImplOptions.AggressiveOptimization | MethodImplOptions.AggressiveInlining)]
        public static StaticModInt<T> Raw(int v)
        {
            var u = unchecked((uint)v);
            Debug.Assert(u < Mod);
            return new StaticModInt<T>(u);
        }

        /// <summary>
        /// StaticModInt&lt;<typeparamref name="T"/>&gt; 型のインスタンスを生成します。
        /// </summary>
        /// <remarks>
        /// <paramref name="v"/>が 0 未満、もしくは mod 以上の場合、自動で mod を取ります。
        /// </remarks>
        public StaticModInt(long v) : this(Round(v)) { }

        [MethodImpl(MethodImplOptions.AggressiveOptimization | MethodImplOptions.AggressiveInlining)]
        private StaticModInt(uint v) => _v = v;

        [MethodImpl(MethodImplOptions.AggressiveOptimization | MethodImplOptions.AggressiveInlining)]
        private static uint Round(long v)
        {
            var x = v % default(T).Mod;
            if (x < 0)
            {
                x += default(T).Mod;
            }
            return (uint)x;
        }

        [MethodImpl(MethodImplOptions.AggressiveOptimization | MethodImplOptions.AggressiveInlining)]
        public static StaticModInt<T> operator ++(StaticModInt<T> value)
        {
            var v = value._v + 1;
            if (v == default(T).Mod)
            {
                v = 0;
            }
            return new StaticModInt<T>(v);
        }

        [MethodImpl(MethodImplOptions.AggressiveOptimization | MethodImplOptions.AggressiveInlining)]
        public static StaticModInt<T> operator --(StaticModInt<T> value)
        {
            var v = value._v;
            if (v == 0)
            {
                v = default(T).Mod;
            }
            return new StaticModInt<T>(v - 1);
        }

        [MethodImpl(MethodImplOptions.AggressiveOptimization | MethodImplOptions.AggressiveInlining)]
        public static StaticModInt<T> operator +(StaticModInt<T> lhs, StaticModInt<T> rhs)
        {
            var v = lhs._v + rhs._v;
            if (v >= default(T).Mod)
            {
                v -= default(T).Mod;
            }
            return new StaticModInt<T>(v);
        }

        [MethodImpl(MethodImplOptions.AggressiveOptimization | MethodImplOptions.AggressiveInlining)]
        public static StaticModInt<T> operator -(StaticModInt<T> lhs, StaticModInt<T> rhs)
        {
            unchecked
            {
                var v = lhs._v - rhs._v;
                if (v >= default(T).Mod)
                {
                    v += default(T).Mod;
                }
                return new StaticModInt<T>(v);
            }
        }

        [MethodImpl(MethodImplOptions.AggressiveOptimization | MethodImplOptions.AggressiveInlining)]
        public static StaticModInt<T> operator *(StaticModInt<T> lhs, StaticModInt<T> rhs)
        {
            return new StaticModInt<T>((uint)((ulong)lhs._v * rhs._v % default(T).Mod));
        }

        /// <summary>
        /// 除算を行います。
        /// </summary>
        /// <remarks>
        /// <para>- 制約: <paramref name="rhs"/> に乗法の逆元が存在する。（gcd(<paramref name="rhs"/>, mod) = 1）</para>
        /// <para>- 計算量: O(log(mod))</para>
        /// </remarks>
        [MethodImpl(MethodImplOptions.AggressiveOptimization | MethodImplOptions.AggressiveInlining)]
        public static StaticModInt<T> operator /(StaticModInt<T> lhs, StaticModInt<T> rhs) => lhs * rhs.Inv();

        public static StaticModInt<T> operator +(StaticModInt<T> value) => value;
        public static StaticModInt<T> operator -(StaticModInt<T> value) => new StaticModInt<T>() - value;
        public static bool operator ==(StaticModInt<T> lhs, StaticModInt<T> rhs) => lhs._v == rhs._v;
        public static bool operator !=(StaticModInt<T> lhs, StaticModInt<T> rhs) => lhs._v != rhs._v;
        public static implicit operator StaticModInt<T>(int value) => new StaticModInt<T>(value);
        public static implicit operator StaticModInt<T>(long value) => new StaticModInt<T>(value);

        /// <summary>
        /// 自身を x として、x^<paramref name="n"/> を返します。
        /// </summary>
        /// <remarks>
        /// <para>制約: 0≤|<paramref name="n"/>|</para>
        /// <para>計算量: O(log(<paramref name="n"/>))</para>
        /// </remarks>
        [MethodImpl(MethodImplOptions.AggressiveOptimization | MethodImplOptions.AggressiveInlining)]
        public StaticModInt<T> Pow(long n)
        {
            Debug.Assert(0 <= n);
            var x = this;
            var r = new StaticModInt<T>(1u);

            while (n > 0)
            {
                if ((n & 1) > 0)
                {
                    r *= x;
                }
                x *= x;
                n >>= 1;
            }

            return r;
        }

        /// <summary>
        /// 自身を x として、 xy≡1 なる y を返します。
        /// </summary>
        /// <remarks>
        /// <para>制約: gcd(x, mod) = 1</para>
        /// </remarks>
        [MethodImpl(MethodImplOptions.AggressiveOptimization | MethodImplOptions.AggressiveInlining)]
        public StaticModInt<T> Inv()
        {
            if (default(T).IsPrime)
            {
                Debug.Assert(_v > 0);
                return Pow(default(T).Mod - 2);
            }
            else
            {
                var (g, x) = Internal.InternalMath.InvGCD(_v, default(T).Mod);
                Debug.Assert(g == 1);
                return new StaticModInt<T>(x);
            }
        }

        public override string ToString() => _v.ToString();
        public override bool Equals(object obj) => obj is StaticModInt<T> && this == (StaticModInt<T>)obj;
        public override int GetHashCode() => _v.GetHashCode();
    }

    /// <summary>
    /// 四則演算時に自動で mod を取る整数型。実行時に mod が決まる場合でも使用可能です。
    /// </summary>
    /// <remarks>
    /// 使用前に DynamicModInt&lt;<typeparamref name="T"/>&gt;.Mod に mod の値を設定する必要があります。
    /// </remarks>
    /// <typeparam name="T">mod の ID を表す構造体</typeparam>
    /// <example>
    /// <code>
    /// using AtCoder.ModInt = AtCoder.DynamicModInt&lt;AtCoder.ModID0&gt;;
    ///
    /// void SomeMethod()
    /// {
    ///     ModInt.Mod = 1000000009;
    ///     var m = new ModInt(1);
    ///     m -= 2;
    ///     Console.WriteLine(m);   // 1000000008
    /// }
    /// </code>
    /// </example>
    public readonly struct DynamicModInt<T> where T : struct, IDynamicModID
    {
        private readonly uint _v;
        private static Internal.Barrett bt;

        /// <summary>
        /// 格納されている値を返します。
        /// </summary>
        public int Value => (int)_v;

        /// <summary>
        /// mod を返します。
        /// </summary>
        public static int Mod
        {
            get => (int)bt.Mod;
            set
            {
                Debug.Assert(1 <= value);
                bt = new Internal.Barrett((uint)value);
            }
        }

        /// <summary>
        /// <paramref name="v"/> に対して mod を取らずに DynamicModInt&lt;<typeparamref name="T"/>&gt; 型のインスタンスを生成します。
        /// </summary>
        /// <remarks>
        /// <para>定数倍高速化のための関数です。 <paramref name="v"/> に 0 未満または mod 以上の値を入れた場合の挙動は未定義です。</para>
        /// <para>制約: 0≤|<paramref name="v"/>|&lt;mod</para>
        /// </remarks>
        public static DynamicModInt<T> Raw(int v)
        {
            var u = unchecked((uint)v);
            Debug.Assert(bt != null, $"使用前に {nameof(DynamicModInt<T>)}<{nameof(T)}>.{nameof(Mod)} プロパティに mod の値を設定してください。");
            Debug.Assert(u < Mod);
            return new DynamicModInt<T>(u);
        }

        /// <summary>
        /// DynamicModInt&lt;<typeparamref name="T"/>&gt; 型のインスタンスを生成します。
        /// </summary>
        /// <remarks>
        /// <para>- 使用前に DynamicModInt&lt;<typeparamref name="T"/>&gt;.Mod に mod の値を設定する必要があります。</para>
        /// <para>- <paramref name="v"/> が 0 未満、もしくは mod 以上の場合、自動で mod を取ります。</para>
        /// </remarks>
        public DynamicModInt(long v) : this(Round(v)) { }

        private DynamicModInt(uint v) => _v = v;

        private static uint Round(long v)
        {
            Debug.Assert(bt != null, $"使用前に {nameof(DynamicModInt<T>)}<{nameof(T)}>.{nameof(Mod)} プロパティに mod の値を設定してください。");
            var x = v % bt.Mod;
            if (x < 0)
            {
                x += bt.Mod;
            }
            return (uint)x;
        }

        public static DynamicModInt<T> operator ++(DynamicModInt<T> value)
        {
            var v = value._v + 1;
            if (v == bt.Mod)
            {
                v = 0;
            }
            return new DynamicModInt<T>(v);
        }

        public static DynamicModInt<T> operator --(DynamicModInt<T> value)
        {
            var v = value._v;
            if (v == 0)
            {
                v = bt.Mod;
            }
            return new DynamicModInt<T>(v - 1);
        }

        public static DynamicModInt<T> operator +(DynamicModInt<T> lhs, DynamicModInt<T> rhs)
        {
            var v = lhs._v + rhs._v;
            if (v >= bt.Mod)
            {
                v -= bt.Mod;
            }
            return new DynamicModInt<T>(v);
        }

        public static DynamicModInt<T> operator -(DynamicModInt<T> lhs, DynamicModInt<T> rhs)
        {
            unchecked
            {
                var v = lhs._v - rhs._v;
                if (v >= bt.Mod)
                {
                    v += bt.Mod;
                }
                return new DynamicModInt<T>(v);
            }
        }

        public static DynamicModInt<T> operator *(DynamicModInt<T> lhs, DynamicModInt<T> rhs)
        {
            uint z = bt.Mul(lhs._v, rhs._v);
            return new DynamicModInt<T>(z);
        }

        /// <summary>
        /// 除算を行います。
        /// </summary>
        /// <remarks>
        /// <para>- 制約: <paramref name="rhs"/> に乗法の逆元が存在する。（gcd(<paramref name="rhs"/>, mod) = 1）</para>
        /// <para>- 計算量: O(log(mod))</para>
        /// </remarks>
        public static DynamicModInt<T> operator /(DynamicModInt<T> lhs, DynamicModInt<T> rhs) => lhs * rhs.Inv();

        public static DynamicModInt<T> operator +(DynamicModInt<T> value) => value;
        public static DynamicModInt<T> operator -(DynamicModInt<T> value) => new DynamicModInt<T>() - value;
        public static bool operator ==(DynamicModInt<T> lhs, DynamicModInt<T> rhs) => lhs._v == rhs._v;
        public static bool operator !=(DynamicModInt<T> lhs, DynamicModInt<T> rhs) => lhs._v != rhs._v;
        public static implicit operator DynamicModInt<T>(int value) => new DynamicModInt<T>(value);
        public static implicit operator DynamicModInt<T>(long value) => new DynamicModInt<T>(value);

        /// <summary>
        /// 自身を x として、x^<paramref name="n"/> を返します。
        /// </summary>
        /// <remarks>
        /// <para>制約: 0≤|<paramref name="n"/>|</para>
        /// <para>計算量: O(log(<paramref name="n"/>))</para>
        /// </remarks>
        public DynamicModInt<T> Pow(long n)
        {
            Debug.Assert(0 <= n);
            var x = this;
            var r = new DynamicModInt<T>(1u);

            while (n > 0)
            {
                if ((n & 1) > 0)
                {
                    r *= x;
                }
                x *= x;
                n >>= 1;
            }

            return r;
        }

        /// <summary>
        /// 自身を x として、 xy≡1 なる y を返します。
        /// </summary>
        /// <remarks>
        /// <para>制約: gcd(x, mod) = 1</para>
        /// </remarks>
        public DynamicModInt<T> Inv()
        {
            var (g, x) = Internal.InternalMath.InvGCD(_v, bt.Mod);
            Debug.Assert(g == 1);
            return new DynamicModInt<T>(x);
        }

        public override string ToString() => _v.ToString();
        public override bool Equals(object obj) => obj is DynamicModInt<T> && this == (DynamicModInt<T>)obj;
        public override int GetHashCode() => _v.GetHashCode();
    }
}


Measured memory_bytes: 55860