結果
| 問題 | No.1307 Rotate and Accumulate |
| ユーザー |
 takytank
|
| 提出日時 | 2020-12-04 01:19:55 |
| 言語 | C#(csc) (csc 3.9.0) |
| 結果 |
AC
|
| 実行時間 | 1,132 ms / 5,000 ms |
| コード長 | 58,829 bytes |
| コンパイル時間 | 3,451 ms |
| コンパイル使用メモリ | 124,160 KB |
| 実行使用メモリ | 36,864 KB |
| 最終ジャッジ日時 | 2024-09-14 11:24:37 |
| 合計ジャッジ時間 | 16,018 ms |
|
ジャッジサーバーID （参考情報） |
judge4 / judge6 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 3 |
| other | AC * 19 |
コンパイルメッセージ
Microsoft (R) Visual C# Compiler version 3.9.0-6.21124.20 (db94f4cc) Copyright (C) Microsoft Corporation. All rights reserved.
ソースコード
using System;using System.Collections;using System.Collections.Generic;using System.Diagnostics;using System.Diagnostics.CodeAnalysis;using System.Globalization;using System.IO;using System.Linq;using System.Numerics;using System.Reflection.Metadata;using System.Runtime.CompilerServices;using System.Runtime.InteropServices;using System.Text;using System.Threading;namespace YukiCoder{public class Program{static void Main(){using var cin = new Scanner();var (n, q) = cin.Int2();var a = cin.ArrayLong(n);var b = new long[n + 2];for (int i = 0; i < q; i++) {int r = cin.Int();b[(n - r) % n]++;}var conv = Math.ConvolutionLong(a, b);var ret = new long[n];for (int i = 0; i < n; i++) {ret[i] += conv[i];ret[i] += conv[i + n];}Console.WriteLine(ret.Join(" "));}}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<typeparamrefname="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 << 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<typeparamrefname="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 << 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<typeparamrefname="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 << 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<typeparamrefname="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 << 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<typeparamrefname="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<typeparamrefname="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 << 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{Butterfly<TMod>.Calculate(a);Butterfly<TMod>.Calculate(b);for (int i = 0; i < a.Length; i++) {a[i] *= b[i];}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) % 2^64const ulong M1M2M3 = Mod1 * Mod2 * Mod3;ulong i1 = (ulong)InternalMath.InvGCD((long)M2M3, (long)Mod1).Item2;ulong i2 = (ulong)InternalMath.InvGCD((long)M1M3, (long)Mod2).Item2;ulong i3 = (ulong)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] - 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.hppx -= 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 << 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 = YukiCoder.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;}}/// <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<AtCoder.Mod1000000007>;////// 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<<typeparamref name="T"/>> 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"/>|<mod</para>/// </remarks>public static StaticModInt<T> Raw(int v){var u = unchecked((uint)v);Debug.Assert(u < Mod);return new StaticModInt<T>(u);}/// <summary>/// StaticModInt<<typeparamref name="T"/>> 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)) { }private StaticModInt(uint v) => _v = v;private static uint Round(long v){var x = v % default(T).Mod;if (x < 0) {x += default(T).Mod;}return (uint)x;}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);}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);}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);}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);}}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>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>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>public StaticModInt<T> Inv(){if (default(T).IsPrime) {Debug.Assert(_v > 0);return Pow(default(T).Mod - 2);} else {var (g, x) = 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<<typeparamref name="T"/>>.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<AtCoder.ModID0>;////// 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 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 Barrett((uint)value);}}/// <summary>/// <paramref name="v"/> に対して mod を取らずに DynamicModInt<<typeparamref name="T"/>> 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"/>|<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<<typeparamref name="T"/>> 型のインスタンスを生成します。/// </summary>/// <remarks>/// <para>- 使用前に DynamicModInt<<typeparamref name="T"/>>.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) = 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();}public static partial class InternalMath{private static readonly Dictionary<int, int> primitiveRootsCache = new Dictionary<int, int>(){{ 2, 1 },{ 167772161, 3 },{ 469762049, 3 },{ 754974721, 11 },{ 998244353, 3 }};/// <summary>/// <paramref name="m"/> の最小の原始根を求めます。/// </summary>/// <remarks>/// 制約: <paramref name="m"/> は素数/// </remarks>public static int PrimitiveRoot(int m){Debug.Assert(m >= 2);if (primitiveRootsCache.TryGetValue(m, out var p)) {return p;}return primitiveRootsCache[m] = 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;}}}}/// <summary>/// <paramref name="x"/>^<paramref name="n"/> mod <paramref name="m"/> を返します。/// </summary>/// <remarks>/// <para>制約: 0≤<paramref name="n"/>, 1≤<paramref name="m"/></para>/// <para>計算量: O(log<paramref name="n"/>)</para>/// </remarks>public static long PowMod(long x, long n, int m){Debug.Assert(0 <= n && 1 <= m);if (m == 1) return 0;Barrett barrett = new Barrett((uint)m);uint r = 1, y = (uint)InternalMath.SafeMod(x, m);while (0 < n) {if ((n & 1) != 0) r = barrett.Mul(r, y);y = barrett.Mul(y, y);n >>= 1;}return r;}public static long SafeMod(long x, long m){x %= m;if (x < 0) x += m;return x;}/// <summary>/// g=gcd(a,b),xa=g(mod b) となるような 0≤x<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;}}/// <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;int ret = 0;int pow = 1;while (n > pow) {++ret;pow *= 2;}return ret;}}/// <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;return (uint)(z % Mod);}}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 = CalcurateSumE();/// <summary>/// sumIE[i] = es[0] * ... * es[i - 1] * ies[i]/// </summary>private static StaticModInt<T>[] sumIE = CalcurateSumIE();public static void Calculate(Span<StaticModInt<T>> a){var n = a.Length;var h = InternalMath.CeilPow2(n);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);for (int i = 0; i < p; i++) {var l = a[i + offset];var r = a[i + offset + p] * now;a[i + offset] = l + r;a[i + offset + p] = l - r;}now *= sumE[InternalBit.BSF(~(uint)s)];}}}public static void CalculateInv(Span<StaticModInt<T>> a){var n = a.Length;var h = InternalMath.CeilPow2(n);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);for (int i = 0; i < p; i++) {var l = a[i + offset];var r = a[i + offset + p];a[i + offset] = l + r;a[i + offset + p] = StaticModInt<T>.Raw(unchecked((int)((ulong)(default(T).Mod + l.Value - r.Value) * (ulong)iNow.Value % default(T).Mod)));}iNow *= sumIE[InternalBit.BSF(~(uint)s)];}}}private static StaticModInt<T>[] CalcurateSumE(){int g = InternalMath.PrimitiveRoot((int)default(T).Mod);int cnt2 = InternalBit.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)) == 1Span<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)) == 1es[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;}private static StaticModInt<T>[] CalcurateSumIE(){int g = InternalMath.PrimitiveRoot((int)default(T).Mod);int cnt2 = InternalBit.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)) == 1Span<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)) == 1es[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;}}public static class InternalBit{/// <summary>/// _blsi_u32 OR <paramref name="n"/> & -<paramref name="n"/>/// <para><paramref name="n"/>で立っているうちの最下位の 1 ビットのみを立てた整数を返す</para>/// </summary>/// <param name="n"></param>/// <returns><paramref name="n"/> & -<paramref name="n"/></returns>[MethodImpl(MethodImplOptions.AggressiveInlining)]public static int ExtractLowestSetBit(int n){return n & -n;}/// <summary>/// (<paramref name="n"/> & (1 << 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>[MethodImpl(MethodImplOptions.AggressiveInlining)]public static int BSF(uint n){for (int i = 0; i < 32; i++) {if (((1 << i) & n) != 0) {return i;}}return 32;}}public struct BitFlag{public static BitFlag Begin() => 0;public static BitFlag End(int bitCount) => 1 << bitCount;public static BitFlag FromBit(int bitNumber) => 1 << bitNumber;private readonly int flags_;public bool this[int bitNumber] => (flags_ & (1 << bitNumber)) != 0;public BitFlag(int flags) { flags_ = flags; }public bool Has(BitFlag target) => (flags_ & target.flags_) == target.flags_;public bool Has(int target) => (flags_ & target) == target;public bool HasBit(int bitNumber) => (flags_ & (1 << bitNumber)) != 0;public BitFlag OrBit(int bitNumber) => (flags_ | (1 << bitNumber));public BitFlag AndBit(int bitNumber) => (flags_ & (1 << bitNumber));public BitFlag XorBit(int bitNumber) => (flags_ ^ (1 << bitNumber));public static BitFlag operator ++(BitFlag src) => new BitFlag(src.flags_ + 1);public static BitFlag operator --(BitFlag src) => new BitFlag(src.flags_ - 1);public static BitFlag operator |(BitFlag lhs, BitFlag rhs)=> new BitFlag(lhs.flags_ | rhs.flags_);public static BitFlag operator |(BitFlag lhs, int rhs)=> new BitFlag(lhs.flags_ | rhs);public static BitFlag operator |(int lhs, BitFlag rhs)=> new BitFlag(lhs | rhs.flags_);public static BitFlag operator &(BitFlag lhs, BitFlag rhs)=> new BitFlag(lhs.flags_ & rhs.flags_);public static BitFlag operator &(BitFlag lhs, int rhs)=> new BitFlag(lhs.flags_ & rhs);public static BitFlag operator &(int lhs, BitFlag rhs)=> new BitFlag(lhs & rhs.flags_);public static bool operator <(BitFlag lhs, BitFlag rhs) => lhs.flags_ < rhs.flags_;public static bool operator <(BitFlag lhs, int rhs) => lhs.flags_ < rhs;public static bool operator <(int lhs, BitFlag rhs) => lhs < rhs.flags_;public static bool operator >(BitFlag lhs, BitFlag rhs) => lhs.flags_ > rhs.flags_;public static bool operator >(BitFlag lhs, int rhs) => lhs.flags_ > rhs;public static bool operator >(int lhs, BitFlag rhs) => lhs > rhs.flags_;public static bool operator <=(BitFlag lhs, BitFlag rhs) => lhs.flags_ <= rhs.flags_;public static bool operator <=(BitFlag lhs, int rhs) => lhs.flags_ <= rhs;public static bool operator <=(int lhs, BitFlag rhs) => lhs <= rhs.flags_;public static bool operator >=(BitFlag lhs, BitFlag rhs) => lhs.flags_ >= rhs.flags_;public static bool operator >=(BitFlag lhs, int rhs) => lhs.flags_ >= rhs;public static bool operator >=(int lhs, BitFlag rhs) => lhs >= rhs.flags_;public static implicit operator BitFlag(int t) => new BitFlag(t);public static implicit operator int(BitFlag t) => t.flags_;//public int PopCount => (int)Popcnt.PopCount((uint)flags_);public override string ToString() => $"{Convert.ToString(flags_, 2).PadLeft(32, '0')} ({flags_})";[MethodImpl(MethodImplOptions.AggressiveInlining)]public void ForEachSubBits(Action<BitFlag> action){for (BitFlag sub = flags_; sub >= 0; --sub) {sub &= flags_;action(sub);}}}public class HashMap<TKey, TValue> : Dictionary<TKey, TValue>{private readonly Func<TKey, TValue> initialzier_;public HashMap(Func<TKey, TValue> initialzier): base(){initialzier_ = initialzier;}public HashMap(Func<TKey, TValue> initialzier, int capacity): base(capacity){initialzier_ = initialzier;}new public TValue this[TKey key]{get{if (TryGetValue(key, out TValue value)) {return value;} else {var init = initialzier_(key);base[key] = init;return init;}}set { base[key] = value; }}public HashMap<TKey, TValue> Merge(HashMap<TKey, TValue> src,Func<TValue, TValue, TValue> mergeValues){foreach (var key in src.Keys) {this[key] = mergeValues(this[key], src[key]);}return this;}}public struct ModInt{//public const long P = 1000000007;public const long P = 998244353;public const long ROOT = 3;// (924844033, 5)// (998244353, 3)// (1012924417, 5)// (167772161, 3)// (469762049, 3)// (1224736769, 3)private long value_;public ModInt(long value)=> value_ = value;public ModInt(long value, bool mods){if (mods) {value %= P;if (value < 0) {value += P;}}value_ = value;}public static ModInt operator +(ModInt lhs, ModInt rhs){lhs.value_ = (lhs.value_ + rhs.value_) % P;return lhs;}public static ModInt operator +(long lhs, ModInt rhs){rhs.value_ = (lhs + rhs.value_) % P;return rhs;}public static ModInt operator +(ModInt lhs, long rhs){lhs.value_ = (lhs.value_ + rhs) % P;return lhs;}public static ModInt operator -(ModInt lhs, ModInt rhs){lhs.value_ = (P + lhs.value_ - rhs.value_) % P;return lhs;}public static ModInt operator -(long lhs, ModInt rhs){rhs.value_ = (P + lhs - rhs.value_) % P;return rhs;}public static ModInt operator -(ModInt lhs, long rhs){lhs.value_ = (P + lhs.value_ - rhs) % P;return lhs;}public static ModInt operator *(ModInt lhs, ModInt rhs){lhs.value_ = lhs.value_ * rhs.value_ % P;return lhs;}public static ModInt operator *(long lhs, ModInt rhs){rhs.value_ = lhs * rhs.value_ % P;return rhs;}public static ModInt operator *(ModInt lhs, long rhs){lhs.value_ = lhs.value_ * rhs % P;return lhs;}public static ModInt operator /(ModInt lhs, ModInt rhs){long exp = P - 2;while (exp > 0) {if (exp % 2 > 0) {lhs *= rhs;}rhs *= rhs;exp /= 2;}return lhs;}public static implicit operator ModInt(long n) => new ModInt(n, true);public static ModInt Inverse(ModInt value) => Pow(value, P - 2);public static ModInt Pow(ModInt value, long k) => Pow(value.value_, k);public static ModInt Pow(long value, long k){long ret = 1;for (k %= P - 1; k > 0; k >>= 1, value = value * value % P) {if ((k & 1) == 1) {ret = ret * value % P;}}return new ModInt(ret);}public static Span<ModInt> NumberTheoreticTransform(Span<int> values, bool inverses = false){var mods = new ModInt[values.Length];for (int i = 0; i < mods.Length; i++) {mods[i] = new ModInt(values[i]);}return NumberTheoreticTransform(mods, inverses);}public static Span<ModInt> NumberTheoreticTransform(Span<long> values, bool inverses = false){var mods = new ModInt[values.Length];for (int i = 0; i < mods.Length; i++) {mods[i] = new ModInt(values[i]);}return NumberTheoreticTransform(mods, inverses);}public static Span<ModInt> NumberTheoreticTransform(Span<ModInt> a, bool inverses = false){int n = a.Length;if (n == 1) {return a;}var b = new ModInt[n].AsSpan();int r = inverses? (int)(P - 1 - (P - 1) / n): (int)((P - 1) / n);ModInt s = Pow(ROOT, r);var kp = new ModInt[n / 2 + 1];kp.AsSpan().Fill(1);for (int i = 0; i < n / 2; ++i) {kp[i + 1] = kp[i] * s;}int l = n / 2;for (int i = 1; i < n; i <<= 1, l >>= 1) {r = 0;for (int j = 0; j < l; ++j, r += i) {s = kp[i * j];for (int k = 0; k < i; ++k) {var p = a[k + r];var q = a[k + r + n / 2];b[k + 2 * r] = p + q;b[k + 2 * r + i] = (p - q) * s;}}var temp = a;a = b;b = temp;}if (inverses) {s = Inverse(n);for (int i = 0; i < n; i++) {a[i] = a[i] * s;}}return a;}public static ModInt[,] NumberTheoreticTransform2D(ModInt[,] a, bool inverses = false){int h = a.GetLength(0);int w = a.GetLength(1);if (h == 1 && w == 1) {return a;}var b = new ModInt[h, w];{int n = w;int r = inverses? (int)(P - 1 - (P - 1) / n): (int)((P - 1) / n);ModInt s = Pow(ROOT, r);var kp = new ModInt[n / 2 + 1];kp.AsSpan().Fill(1);for (int i = 0; i < n / 2; ++i) {kp[i + 1] = kp[i] * s;}for (int y = 0; y < h; y++) {int l = n / 2;for (int i = 1; i < n; i <<= 1, l >>= 1) {r = 0;for (int j = 0; j < l; ++j, r += i) {s = kp[i * j];for (int k = 0; k < i; ++k) {var p = a[y, k + r];var q = a[y, k + r + n / 2];b[y, k + 2 * r] = p + q;b[y, k + 2 * r + i] = (p - q) * s;}}var temp = a;a = b;b = temp;}if (inverses) {s = Inverse(n);for (int i = 0; i < n; i++) {a[y, i] = a[y, i] * s;}}}}for (int i = 0; i < h; i++) {for (int j = 0; j < w; j++) {b[h, w] = 0;}}{int n = h;int r = inverses? (int)(P - 1 - (P - 1) / n): (int)((P - 1) / n);ModInt s = Pow(ROOT, r);var kp = new ModInt[n / 2 + 1];kp.AsSpan().Fill(1);for (int i = 0; i < n / 2; ++i) {kp[i + 1] = kp[i] * s;}for (int x = 0; x < w; x++) {int l = n / 2;for (int i = 1; i < n; i <<= 1, l >>= 1) {r = 0;for (int j = 0; j < l; ++j, r += i) {s = kp[i * j];for (int k = 0; k < i; ++k) {var p = a[k + r, x];var q = a[k + r + n / 2, x];b[k + 2 * r, x] = p + q;b[k + 2 * r + i, x] = (p - q) * s;}}var temp = a;a = b;b = temp;}if (inverses) {s = Inverse(n);for (int i = 0; i < n; i++) {a[i, x] = a[i, x] * s;}}}}return a;}public static Span<ModInt> Convolve(ReadOnlySpan<ModInt> a, ReadOnlySpan<ModInt> b){int resultLength = a.Length + b.Length - 1;int nttLenght = 1;while (nttLenght < resultLength) {nttLenght <<= 1;}var aa = new ModInt[nttLenght];a.CopyTo(aa);var bb = new ModInt[nttLenght];b.CopyTo(bb);var fa = NumberTheoreticTransform(aa);var fb = NumberTheoreticTransform(bb);for (int i = 0; i < nttLenght; i++) {fa[i] *= fb[i];}var convolved = NumberTheoreticTransform(fa, true);return convolved.Slice(0, resultLength);}public long ToLong() => value_;public override string ToString() => value_.ToString();}public static class Helper{[MethodImpl(MethodImplOptions.AggressiveInlining)]public static void UpdateMin<T>(this ref T target, T value) where T : struct, IComparable<T>=> target = target.CompareTo(value) > 0 ? value : target;[MethodImpl(MethodImplOptions.AggressiveInlining)]public static void UpdateMin<T>(this ref T target, T value, Action<T> onUpdated)where T : struct, IComparable<T>{if (target.CompareTo(value) > 0) {target = value;onUpdated(value);}}[MethodImpl(MethodImplOptions.AggressiveInlining)]public static void UpdateMax<T>(this ref T target, T value) where T : struct, IComparable<T>=> target = target.CompareTo(value) < 0 ? value : target;[MethodImpl(MethodImplOptions.AggressiveInlining)]public static void UpdateMax<T>(this ref T target, T value, Action<T> onUpdated)where T : struct, IComparable<T>{if (target.CompareTo(value) < 0) {target = value;onUpdated(value);}}[MethodImpl(MethodImplOptions.AggressiveInlining)]public static T[] Array1<T>(int n, T initialValue) where T : struct=> new T[n].Fill(initialValue);[MethodImpl(MethodImplOptions.AggressiveInlining)]public static T[] Array1<T>(int n, Func<int, T> initializer)=> Enumerable.Range(0, n).Select(x => initializer(x)).ToArray();[MethodImpl(MethodImplOptions.AggressiveInlining)]public static T[] Fill<T>(this T[] array, T value)where T : struct{array.AsSpan().Fill(value);return array;}[MethodImpl(MethodImplOptions.AggressiveInlining)]public static T[,] Array2<T>(int n, int m, T initialValule) where T : struct=> new T[n, m].Fill(initialValule);[MethodImpl(MethodImplOptions.AggressiveInlining)]public static T[,] Array2<T>(int n, int m, Func<int, int, T> initializer){var array = new T[n, m];for (int i = 0; i < n; ++i) {for (int j = 0; j < m; ++j) {array[i, j] = initializer(i, j);}}return array;}[MethodImpl(MethodImplOptions.AggressiveInlining)]public static T[,] Fill<T>(this T[,] array, T initialValue)where T : struct{MemoryMarshal.CreateSpan<T>(ref array[0, 0], array.Length).Fill(initialValue);return array;}[MethodImpl(MethodImplOptions.AggressiveInlining)]public static Span<T> AsSpan<T>(this T[,] array, int i)=> MemoryMarshal.CreateSpan<T>(ref array[i, 0], array.GetLength(1));[MethodImpl(MethodImplOptions.AggressiveInlining)]public static T[,,] Array3<T>(int n1, int n2, int n3, T initialValue)where T : struct=> new T[n1, n2, n3].Fill(initialValue);[MethodImpl(MethodImplOptions.AggressiveInlining)]public static T[,,] Fill<T>(this T[,,] array, T initialValue)where T : struct{MemoryMarshal.CreateSpan<T>(ref array[0, 0, 0], array.Length).Fill(initialValue);return array;}[MethodImpl(MethodImplOptions.AggressiveInlining)]public static Span<T> AsSpan<T>(this T[,,] array, int i, int j)=> MemoryMarshal.CreateSpan<T>(ref array[i, j, 0], array.GetLength(2));[MethodImpl(MethodImplOptions.AggressiveInlining)]public static T[,,,] Array4<T>(int n1, int n2, int n3, int n4, T initialValue)where T : struct=> new T[n1, n2, n3, n4].Fill(initialValue);[MethodImpl(MethodImplOptions.AggressiveInlining)]public static T[,,,] Fill<T>(this T[,,,] array, T initialValue)where T : struct{MemoryMarshal.CreateSpan<T>(ref array[0, 0, 0, 0], array.Length).Fill(initialValue);return array;}[MethodImpl(MethodImplOptions.AggressiveInlining)]public static Span<T> AsSpan<T>(this T[,,,] array, int i, int j, int k)=> MemoryMarshal.CreateSpan<T>(ref array[i, j, k, 0], array.GetLength(3));private static readonly int[] delta4_ = { 1, 0, -1, 0, 1 };[MethodImpl(MethodImplOptions.AggressiveInlining)]public static void DoIn4(int i, int j, int imax, int jmax, Action<int, int> action){for (int dn = 0; dn < 4; ++dn) {int d4i = i + delta4_[dn];int d4j = j + delta4_[dn + 1];if ((uint)d4i < (uint)imax && (uint)d4j < (uint)jmax) {action(d4i, d4j);}}}private static readonly int[] delta8_ = { 1, 0, -1, 0, 1, 1, -1, -1, 1 };[MethodImpl(MethodImplOptions.AggressiveInlining)]public static void DoIn8(int i, int j, int imax, int jmax, Action<int, int> action){for (int dn = 0; dn < 8; ++dn) {int d8i = i + delta8_[dn];int d8j = j + delta8_[dn + 1];if ((uint)d8i < (uint)imax && (uint)d8j < (uint)jmax) {action(d8i, d8j);}}}[MethodImpl(MethodImplOptions.AggressiveInlining)]public static void ForEachSubBits(int bit, Action<int> action){for (int sub = bit; sub >= 0; --sub) {sub &= bit;action(sub);}}[MethodImpl(MethodImplOptions.AggressiveInlining)]public static string Reverse(string src){var chars = src.ToCharArray();for (int i = 0, j = chars.Length - 1; i < j; ++i, --j) {var tmp = chars[i];chars[i] = chars[j];chars[j] = tmp;}return new string(chars);}[MethodImpl(MethodImplOptions.AggressiveInlining)]public static string Join<T>(this IEnumerable<T> values, string separator = "")=> string.Join(separator, values);[MethodImpl(MethodImplOptions.AggressiveInlining)]public static string JoinNL<T>(this IEnumerable<T> values)=> string.Join(Environment.NewLine, values);}public class Scanner : IDisposable{private const int BUFFER_SIZE = 1024;private const int ASCII_CHAR_BEGIN = 33;private const int ASCII_CHAR_END = 126;private readonly string filePath_;private readonly Stream stream_;private readonly byte[] buf_ = new byte[BUFFER_SIZE];private int length_ = 0;private int index_ = 0;private bool isEof_ = false;public Scanner(string file = ""){if (string.IsNullOrWhiteSpace(file)) {stream_ = Console.OpenStandardInput();} else {filePath_ = file;stream_ = new FileStream(file, FileMode.Open);}Console.SetOut(new StreamWriter(Console.OpenStandardOutput()) {AutoFlush = false});}public void Dispose(){Console.Out.Flush();stream_.Dispose();}[MethodImpl(MethodImplOptions.AggressiveInlining)]public char Char(){byte b;do {b = Read();} while (b < ASCII_CHAR_BEGIN || ASCII_CHAR_END < b);return (char)b;}[MethodImpl(MethodImplOptions.AggressiveInlining)]public string Next(){var sb = new StringBuilder();for (var b = Char(); b >= ASCII_CHAR_BEGIN && b <= ASCII_CHAR_END; b = (char)Read()) {sb.Append(b);}return sb.ToString();}[MethodImpl(MethodImplOptions.AggressiveInlining)]public string[] ArrayString(int length){var array = new string[length];for (int i = 0; i < length; ++i) {array[i] = Next();}return array;}[MethodImpl(MethodImplOptions.AggressiveInlining)]public int Int() => (int)Long();[MethodImpl(MethodImplOptions.AggressiveInlining)]public int Int(int offset) => Int() + offset;[MethodImpl(MethodImplOptions.AggressiveInlining)]public (int, int) Int2(int offset = 0)=> (Int(offset), Int(offset));[MethodImpl(MethodImplOptions.AggressiveInlining)]public (int, int, int) Int3(int offset = 0)=> (Int(offset), Int(offset), Int(offset));[MethodImpl(MethodImplOptions.AggressiveInlining)]public (int, int, int, int) Int4(int offset = 0)=> (Int(offset), Int(offset), Int(offset), Int(offset));[MethodImpl(MethodImplOptions.AggressiveInlining)]public int[] ArrayInt(int length, int offset = 0){var array = new int[length];for (int i = 0; i < length; ++i) {array[i] = Int(offset);}return array;}[MethodImpl(MethodImplOptions.AggressiveInlining)]public long Long(){long ret = 0;byte b;bool ng = false;do {b = Read();} while (b != '-' && (b < '0' || '9' < b));if (b == '-') {ng = true;b = Read();}for (; true; b = Read()) {if (b < '0' || '9' < b) {return ng ? -ret : ret;} else {ret = ret * 10 + b - '0';}}}[MethodImpl(MethodImplOptions.AggressiveInlining)]public long Long(long offset) => Long() + offset;[MethodImpl(MethodImplOptions.AggressiveInlining)]public (long, long) Long2(long offset = 0)=> (Long(offset), Long(offset));[MethodImpl(MethodImplOptions.AggressiveInlining)]public (long, long, long) Long3(long offset = 0)=> (Long(offset), Long(offset), Long(offset));[MethodImpl(MethodImplOptions.AggressiveInlining)]public (long, long, long, long) Long4(long offset = 0)=> (Long(offset), Long(offset), Long(offset), Long(offset));[MethodImpl(MethodImplOptions.AggressiveInlining)]public long[] ArrayLong(int length, long offset = 0){var array = new long[length];for (int i = 0; i < length; ++i) {array[i] = Long(offset);}return array;}[MethodImpl(MethodImplOptions.AggressiveInlining)]public BigInteger Big() => new BigInteger(Long());[MethodImpl(MethodImplOptions.AggressiveInlining)]public BigInteger Big(long offset) => Big() + offset;[MethodImpl(MethodImplOptions.AggressiveInlining)]public (BigInteger, BigInteger) Big2(long offset = 0)=> (Big(offset), Big(offset));[MethodImpl(MethodImplOptions.AggressiveInlining)]public (BigInteger, BigInteger, BigInteger) Big3(long offset = 0)=> (Big(offset), Big(offset), Big(offset));[MethodImpl(MethodImplOptions.AggressiveInlining)]public (BigInteger, BigInteger, BigInteger, BigInteger) Big4(long offset = 0)=> (Big(offset), Big(offset), Big(offset), Big(offset));[MethodImpl(MethodImplOptions.AggressiveInlining)]public BigInteger[] ArrayBig(int length, long offset = 0){var array = new BigInteger[length];for (int i = 0; i < length; ++i) {array[i] = Big(offset);}return array;}[MethodImpl(MethodImplOptions.AggressiveInlining)]public double Double() => double.Parse(Next(), CultureInfo.InvariantCulture);[MethodImpl(MethodImplOptions.AggressiveInlining)]public double Double(double offset) => Double() + offset;[MethodImpl(MethodImplOptions.AggressiveInlining)]public (double, double) Double2(double offset = 0)=> (Double(offset), Double(offset));[MethodImpl(MethodImplOptions.AggressiveInlining)]public (double, double, double) Double3(double offset = 0)=> (Double(offset), Double(offset), Double(offset));[MethodImpl(MethodImplOptions.AggressiveInlining)]public (double, double, double, double) Double4(double offset = 0)=> (Double(offset), Double(offset), Double(offset), Double(offset));[MethodImpl(MethodImplOptions.AggressiveInlining)]public double[] ArrayDouble(int length, double offset = 0){var array = new double[length];for (int i = 0; i < length; ++i) {array[i] = Double(offset);}return array;}private byte Read(){if (isEof_) {throw new EndOfStreamException();}if (index_ >= length_) {index_ = 0;if ((length_ = stream_.Read(buf_, 0, BUFFER_SIZE)) <= 0) {isEof_ = true;return 0;}}return buf_[index_++];}public void Save(string text){if (string.IsNullOrWhiteSpace(filePath_)) {return;}File.WriteAllText(filePath_ + "_output.txt", text);}}}