#588938 (C#(csc)) No.1307 Rotate and Accumulate

提出ソース

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

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テストケース

テストケース表示

入力	結果	実行時間実行使用メモリ
testcase_00	AC	38 ms 18,176 KB
testcase_01	AC	38 ms 18,176 KB
testcase_02	AC	38 ms 18,176 KB
testcase_03	AC	39 ms 18,176 KB
testcase_04	AC	39 ms 18,304 KB
testcase_05	AC	39 ms 18,176 KB
testcase_06	AC	38 ms 18,176 KB
testcase_07	AC	38 ms 18,176 KB
testcase_08	AC	1,096 ms 34,304 KB
testcase_09	AC	1,101 ms 34,944 KB
testcase_10	AC	560 ms 28,032 KB
testcase_11	AC	566 ms 30,848 KB
testcase_12	AC	561 ms 28,416 KB
testcase_13	AC	99 ms 18,560 KB
testcase_14	AC	288 ms 22,528 KB
testcase_15	AC	1,132 ms 36,864 KB
testcase_16	AC	1,130 ms 36,736 KB
testcase_17	AC	1,132 ms 36,736 KB
testcase_18	AC	914 ms 36,480 KB
testcase_19	AC	977 ms 36,608 KB
testcase_20	AC	926 ms 36,736 KB
testcase_21	AC	38 ms 17,920 KB

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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

raw source code

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<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 << 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 << 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 << 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 << 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 << 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) % 2^64
				const 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.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 << 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&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>
		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)) { }

		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&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 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&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) = 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&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;
			}
		}

		/// <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)) == 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;
		}

		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)) == 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;
		}
	}

	public static class InternalBit
	{
		/// <summary>
		/// _blsi_u32 OR <paramref name="n"/> &amp; -<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"/> &amp; -<paramref name="n"/></returns>
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		public static int ExtractLowestSetBit(int n)
		{
			return n & -n;
		}

		/// <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>
		[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);
		}
	}
}

yukicoder

結果

テストケース

コンパイルメッセージ

ソースコード