Problem:
Score : 800 points

Problem Statement

Given are integers N and X. For each integer k between 0 and X (inclusive), find the answer to the following question, then compute the sum of all those answers, modulo 998244353.

Let us repeat the following operation on the integer k. Operation: if the integer is currently odd, subtract 1 from it and divide it by 2; otherwise, divide it by 2 and add 2^{N-1} to it. How many operations need to be performed until k returns to its original value? (The answer is considered to be 0 if k never returns to its original value.)

Constraints

1 \leq N \leq 2\times 10^5

0 \leq X < 2^N

X is given in binary and has exactly N digits. (In case X has less than N digits, it is given with leading zeroes.)

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
X

Output

Print the sum of the answers to the questions for the integers between 0 and X (inclusive), modulo 998244353.

Sample Input 1

3
111

Sample Output 1

40

For example, for k=3, the operation changes k as follows: 1,0,4,6,7,3. Therefore the answer for k=3 is 6.

Sample Input 2

6
110101

Sample Output 2

616

Sample Input 3

30
001110011011011101010111011100

Sample Output 3

549320998

Solution in Java8 (OpenJDK 1.8.0) (judge verdict: Wrong Answer)



import java.awt.Point;
import java.io.IOException;
import java.io.InputStream;
import java.io.PrintStream;
import java.io.PrintWriter;
import java.io.Serializable;
import java.math.RoundingMode;
import java.util.AbstractList;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.Collection;
import java.util.Collections;
import java.util.Comparator;
import java.util.HashSet;
import java.util.Iterator;
import java.util.List;
import java.util.Locale;
import java.util.Map;
import java.util.NoSuchElementException;
import java.util.RandomAccess;
import java.util.TreeMap;
import java.util.function.BinaryOperator;
import java.util.function.UnaryOperator;

public class Main implements Runnable{

	private void solve(FastIO io) {
		/*
		 * author: 31536000
		 * AGC039 C問題
		 * 考察メモ
		 * 1. 0...01...1や1...10...0のような数について
		 * これは2N回で循環することが分かるので2N
		 * 次に、0...01..10..0について
		 * これはまず最初の0全部が1全部になって後半に動くので1..10...01..1になり、次に……とやっぱり循環する
		 * というかどんな数でも2N回以内に循環するね
		 * 問題は、その循環が2Nより短い数があるということ
		 * 例えば010は010→101→010になる
		 * これはどういう規則で出るのか？
		 * 0100→1010→1101→0110→1011→0101→0010→1001→0100、これは8回だね
		 * 01010→10101→01010、これは0回→対称性、ではないね(1001とかが嘘)
		 * 1001→0100→1010→1101→0110→…で8回
		 * ということで0101…の組み合わせだけ2回で終わる、と考える
		 * これは示せて、01...01→10...10→01...01なのでね
		 * 同様に1010…も同じ
		 * 他の組み合わせもないだろうか
		 * 大事な性質として、後ろk個を選ぶとその01が反転したものが後ろにくっつく
		 * ということは、その時点で終わるとしたら前からk個のreverseが後ろと同値、かつそれ以外が移行したものと同値
		 * ん、数列が帰納的に定義できて、末尾kの数列をAとする
		 * とすると移動した先、次のk個もAでないと矛盾
		 * 更に次のk個もAでないと矛盾
		 * ……と考えていくと、最終的に全てがAになるね
		 * しかもAの反転はA、ということは求めるべきはbit反転して逆向きにしたものが元の数列に一致するようなもの
		 * これの繰り返し
		 * まずNの約数以外は無視して良い、それ以外は成立しえない
		 * Nの約数でこれを考えればよくて
		 * よって、後はこれを各Nの約数について高速に数え上げられれば答えがほぼ求まったも同然
		 * 01、0011、000111……みたいなのは満たしそうだけど、それ以外にあるかなぁ？
		 * 010101とか011001も良いのか、要するに前半半分の集合を決めると後半半分はその逆の位置の決め方で決まるので
		 * ということは、奇数だと無い、も言えるのか……？
		 * これは嘘っぽくて、01010が反例になってるけども
		 * これも01の繰り返しで、周期2なのよね
		 * あれ、約数という前提崩れてないか、再検討だー！
		 * 001100110を考えると001100110→100110011→010011001→001001100→100100110→110010011→……えーと周期18
		 * さぁ、01010との違いは何だった？
		 * 左右両方から見てこの周期数列だとうまくいく、とか？
		 * これ正しそう、左右両方からみて周期kが存在すると矛盾なく動きそうね
		 * さて、そのようなものってどう求める……？
		 * そもそも、N=xk+yとして、左右にyのはみだしが発生するわけだ(さっきのだと9=2k+1なので両方に1のはみだし)
		 * この時、y=1だと0101...の2パターンのみ
		 * y=2だとxが2の倍数なら4パターン、そうでないなら2パターン
		 * 何が起こったかというと、gcd(x,y)で決まる
		 * つまり、長さkの周期列が存在する→kがNの約数である、で問題無かった
		 * 後は重複排除を良い感じにしたい、例えば長さ4の列として0101(周期2)は勘弁したい
		 * 長さ6の列として010101もヤバくて、つまり更にkの約数で作られるとヤバそう
		 * ところでこれは包除原理なので
		 * 下位bitから決め打つと良さそうね
		 *
		 * 更に注意点
		 * さっきのgcd、適当に考えすぎたのでもう少し考える
		 * N=9,k=6だと121[1'2'1'121]で4通り、N=12,k=8だと1221[1'2'2'1'1221]とかになる
		 * つまり、2^((gcd(k,N%k)+1)/2)かなぁ
		 * 後は各kについて考えていけば良さそう
		 * ちなみにkが奇数は存在しないです(中央bitが矛盾なので)
		 *
		 * まず、kについて全部ループを回すことを考えるO(N)
		 * ここで、kについて2^((gcd(k,N%k)+1)/2)通りの答えがkになるようなパターンがあることが分かる
		 * ただ、これがXを超えるか判定する必要がある(ねぇ、このパート必要だった？)
		 * 勿論この類は桁DPなので……
		 * 先頭のgcd(k, N%k)桁について実際に作ってみることを考える
		 * 約数の個数っぽいのでO(N√N)なら間に合うでしょって思っておこう
		 *
		 *とかいろいろ考察してたけど更にシンプルにできるね
		 *詳しくはコードの方で
		 *
		 */
		int N = io.nextInt();
		char[] X = io.next().toCharArray();
		ModUtility mod = new ModUtility(998244353);
		ModInteger[] Xdigit = new ModInteger[N + 1];
		Xdigit[0] = mod.create(0);
		for (int i = 0;i < N;++ i) {
			Xdigit[i + 1] = Xdigit[i].multiply(2);
			if (X[i] == '1') Xdigit[i + 1].addEqual(1);
		}
		ModInteger ans = mod.create(1);
		ans.addEqual(Xdigit[N]); // 0以上X以下、なのでX+1個ある
		ans.multiplyEqual(2 * N); // これが周期が存在しないと仮定したときの答え
		ModInteger[] calc = new ModInteger[N + 1];
		for (int i = 1;i < N / 2;++ i) {
			if (IntegerMath.gcd(N, 2 * i) == i) { // このパターンのみが構築可能
				calc[i] = Xdigit[i].add(1);
				for (int j = 2;j * j <= i;++ j) {
					if (i % j == 0 && calc[j] != null) calc[i].subtractEqual(calc[j]); // 重複の削除
				}
				for (int j = i;j < N;++ j) { // 最初i桁が同値のものだけ、本当に作れるか検証が必要
					char num;
					if (j % (2 * i) < i) num = X[j % i];
					else num = (char)(1 - X[i - j % i - 1] + '0' + '0');
					if (num != X[j]) {
						calc[i].subtractEqual(num - '0');
						break;
					}
				}
				ans.subtractEqual(calc[i].multiply(N - i).multiply(2));
			}
		}
		io.println(ans);

	}

	/** デバッグ用コードのお供に */
	private static boolean DEBUG = false;
	/** 確保するメモリの大きさ(単位: MB)*/
	private static final long MEMORY = 64;
	private final FastIO io;

	public static void main(String[] args) {
	        Thread.setDefaultUncaughtExceptionHandler((t, e) -> e.printStackTrace());
	        new Thread(null, new Main(), "", MEMORY * 1048576).start();
	}

	public Main() {
		io = new FastIO();
		assert (DEBUG = true) | true; // yukicoderだと-eaが付いてるので正しく動かないことに注意
		if (DEBUG) {
			io.setAutoFlush(true);
			io.println("debug mode");
		}
	}

	@Override
	public void run() {
		solve(io);
		io.flush();
	}

	// 以下、ライブラリ

	/**
	 * int型に対する様々な演算を提供します。
	 * @author 31536000
	 *
	 */
	public static class IntegerMath {
		private IntegerMath() {
			throw new AssertionError();
		}


		/**
		 * aとbのうち、小さい方を返します。
		 * @param a 整数
		 * @param b 整数
		 * @return aとbのうち小さい方の値
		 */
		public static int min(int a, int b) {
			return a < b ? a : b;
		}

		/**
		 * 配列の中で最小の値を返します。
		 * @param array 配列
		 * @return 配列の中で最小の値
		 */
		public static int min(int... array) {
			int ret = array[0];
			for (int i = 1;i < array.length;++ i) ret = min(ret, array[i]);
			return ret;
		}

		/**
		 * aとbのうち、大きい方を返します。
		 * @param a 整数
		 * @param b 整数
		 * @return aとbのうち大きい方の値
		 */
		public static int max(int a, int b) {
			return a > b ? a : b;
		}

		/**
		 * 配列の中で最大の値を返します。
		 * @param array 配列
		 * @return 配列の中で最大の値
		 */
		public static int max(int... array) {
			int ret = array[0];
			for (int i = 1;i < array.length;++ i) ret = max(ret, array[i]);
			return ret;
		}

		/**
		 * aとbの最大公約数を返します。
		 * @param a 整数
		 * @param b 整数
		 * @return 最大公約数
		 */
		public static int gcd(int a, int b) {
			while (a != 0) if ((b %= a) != 0) a %= b; else return a;
			return b;
		}

		/**
		 * 配列全ての値の最大公約数を返します。
		 * @param array 配列
		 * @return 最大公約数
		 */
		public static int gcd(int... array) {
			int ret = array[0];
			for (int i = 1;i < array.length;++ i) ret = gcd(ret, array[i]);
			return ret;
		}

		/**
		 * 配列全ての値の最小公倍数を返します。
		 * @param a 整数
		 * @param b 整数
		 * @return 最小公倍数
		 */
		public static int lcm(int a, int b) {
			return a / gcd(a, b) * b;
		}

		/**
		 * 配列全ての値の最小公倍数を返します。
		 * @param array 配列
		 * @return 最小公倍数
		 */
		public static int lcm(int... array) {
			int ret = array[0];
			for (int i = 1;i < array.length;++ i) ret = lcm(ret, array[i]);
			return ret;
		}

		/**
		 * aのb乗を返します。
		 * @param a 整数
		 * @param b 整数
		 * @return aのb乗
		 */
		public static int pow(int a, int b) {
			int ans = 1;
			for (int mul = a;b > 0;b >>= 1, mul *= mul) if ((b & 1) != 0) ans *= mul;
			return ans;
		}

		/**
		 * aのb乗をmodを法として計算したものを返します。
		 * @param a 整数
		 * @param b 整数
		 * @param mod 法
		 * @return aのb乗をmodを法として計算したもの
		 */
		public static int pow(int a, int b, int mod) {
			if (b < 0) b = b % (mod - 1) + mod - 1;
			long ans = 1;
			for (long mul = a;b > 0;b >>= 1, mul = mul * mul % mod) if ((b & 1) != 0) ans = ans * mul % mod;
			return (int)ans;
		}

		/**
		 * aをbで割った値を、端数に関してはRoundingModeに従って計算します。
		 * @param a 整数
		 * @param b 整数
		 * @param mode 端数の処理方法
		 * @return a/b
		 * @throws ArithmeticException 不正引数
		 */
		public static int divide(int a, int b, RoundingMode mode) {
			int t;
			switch (mode) {
				case UP:
					t = (a >>> 31 << 1) - 1;
					return (a - t) / b + t;
				case DOWN:
					return a / b;
				case CEILING:
					t = ~a >>> 31;
					return (a - t) / b + t;
				case FLOOR:
					t = a >>> 31;
					return (a + t) / b - t;
				case HALF_UP:
					t = a % b;
					break;
				case HALF_EVEN:
					break;
				case HALF_DOWN:
					break;
				case UNNECESSARY:
					throw new ArithmeticException();
			}
			throw new AssertionError("undefined mode");
		}

		public static int sum(int... array) {
			int ret = 0;
			for (int i : array) ret += i;
			return ret;
		}

		public static int multiply(int... array) {
			int ret = 0;
			for (int i : array) ret *= i;
			return ret;
		}

		public static int nextMultiple(int a, int b) {
			return divide(a, b, RoundingMode.CEILING) * b;
		}

		public static int mod(int a, int mod) {
			return (a %= mod) < 0 ? a + mod : a;
		}

		public enum Addition implements Abelian<Integer>{
			INSTANCE;

			@Override
			public Integer apply(Integer left, Integer right) {
				return left + right;
			}

			@Override
			public Integer unit() {
				return 0;
			}

			@Override
			public Integer inverse(Integer element) {
				return -element;
			}

			@Override
			public Integer hyper(Integer element, int repeat) {
				return element * repeat;
			}
		}

		public enum Multiplication implements CommutativeMonoid<Integer>{
			INSTANCE;

			@Override
			public Integer apply(Integer left, Integer right) {
				return left * right;
			}

			@Override
			public Integer unit() {
				return 1;
			}

			@Override
			public Integer hyper(Integer element, int repeat) {
				return IntegerMath.pow(element, repeat);
			}
		}

		public enum OR implements CommutativeMonoid<Integer> {
			INSTANCE;

			@Override
			public Integer apply(Integer left, Integer right) {
				return left | right;
			}

			@Override
			public Integer unit() {
				return 0;
			}

			@Override
			public Integer hyper(Integer element, int repeat) {
				return element;
			}
		}

		public enum AND implements CommutativeMonoid<Integer> {
			INSTANCE;

			@Override
			public Integer apply(Integer left, Integer right) {
				return left & right;
			}

			@Override
			public Integer unit() {
				return 1;
			}

			@Override
			public Integer hyper(Integer element, int repeat) {
				return element;
			}
		}

		public enum XOR implements Abelian<Integer> {
			INSTANCE;

			@Override
			public Integer apply(Integer left, Integer right) {
				return left ^ right;
			}

			@Override
			public Integer unit() {
				return 0;
			}

			@Override
			public Integer inverse(Integer element) {
				return element;
			}

			@Override
			public Integer hyper(Integer element, int repeat) {
				return (repeat & 1) == 0 ? 0 : element;
			}
		}

		public Abelian<Integer> getAddition() {
			return Addition.INSTANCE;
		}

		public CommutativeMonoid<Integer> getMultiplication() {
			return Multiplication.INSTANCE;
		}

		public CommutativeMonoid<Integer> getOR() {
			return OR.INSTANCE;
		}

		public CommutativeMonoid<Integer> getAND() {
			return AND.INSTANCE;
		}

		public Abelian<Integer> getXOR() {
			return XOR.INSTANCE;
		}

	}

	public static class FastIO {
		private final InputStream in;
		private final byte[] buffer = new byte[1024];
		private int read = 0;
		private int length = 0;
		public final PrintWriter out;
		public final PrintWriter err;
		private boolean autoFlush = false;

		public FastIO() {
			this(System.in, System.out, System.err);
		}

		public FastIO(InputStream in, PrintStream out, PrintStream err) {
			this.in = in;
			this.out = new PrintWriter(out, false);
			this.err = new PrintWriter(err, false);
		}

		public void setAutoFlush(boolean flush) {
			autoFlush = flush;
		}

		private boolean hasNextByte() {
			if (read < length) return true;
			read = 0;
			try {
				length = in.read(buffer);
			} catch (IOException e) {
				e.printStackTrace();
			}
			return length > 0;
		}

		private int readByte() {
			return hasNextByte() ? buffer[read++] : -1;
		}

		private static boolean isPrintableChar(int c) {
			return 33 <= c && c <= 126;
		}

		private static boolean isNumber(int c) {
			return '0' <= c && c <= '9';
		}

		public boolean hasNext() {
			while (hasNextByte() && !isPrintableChar(buffer[read])) read++;
			return hasNextByte();
		}

		public char nextChar() {
			if (!hasNextByte())  throw new NoSuchElementException();
			return (char)readByte();
		}

		public char[][] nextChar(int height) {
			char[][] ret = new char[height][];
			for (int i = 0;i < ret.length;++ i) ret[i] = next().toCharArray();
			return ret;
		}

		public String next() {
			if (!hasNext()) throw new NoSuchElementException();
			StringBuilder sb = new StringBuilder();
			int b;
			while (isPrintableChar(b = readByte())) sb.appendCodePoint(b);
			return sb.toString();
		}

		public String nextLine() {
			StringBuilder sb = new StringBuilder();
			int b;
			while(!isPrintableChar(b = readByte()));
			do sb.appendCodePoint(b); while(isPrintableChar(b = readByte()) || b == ' ');
			return sb.toString();
		}

		public long nextLong() {
			if (!hasNext()) throw new NoSuchElementException();
			long n = 0;
			boolean minus = false;
			int b = readByte();
			if (b == '-') {
				minus = true;
				b = readByte();
			}
			if (!isNumber(b)) throw new NumberFormatException();
			while (true) {
				if (isNumber(b)) {
					n *= 10;
					n += b - '0';
				} else if (b == -1 || !isPrintableChar(b)) return minus ? -n : n;
				else throw new NumberFormatException();
				b = readByte();
			}
		}

		public int nextInt() {
			long nl = nextLong();
			if (nl < Integer.MIN_VALUE || nl > Integer.MAX_VALUE) throw new NumberFormatException();
			return (int) nl;
		}

		public double nextDouble() {
			return Double.parseDouble(next());
		}

		public int[] nextInt(int width) {
			int[] ret = new int[width];
			for (int i = 0;i < width;++ i) ret[i] = nextInt();
			return ret;
		}

		public int[] nextInts() {
			return nextInts(" ");
		}

		public int[] nextInts(String parse) {
			String[] get = nextLine().split(parse);
			int[] ret = new int[get.length];
			for (int i = 0;i < ret.length;++ i) ret[i] = Integer.valueOf(get[i]);
			return ret;
		}

		public long[] nextLong(int width) {
			long[] ret = new long[width];
			for (int i = 0;i < width;++ i) ret[i] = nextLong();
			return ret;
		}

		public long[] nextLongs() {
			return nextLongs(" ");
		}

		public long[] nextLongs(String parse) {
			String[] get = nextLine().split(parse);
			long[] ret = new long[get.length];
			for (int i = 0;i < ret.length;++ i) ret[i] = Long.valueOf(get[i]);
			return ret;
		}

		public int[][] nextInt(int width, int height) {
			int[][] ret = new int[height][width];
			for (int i = 0, j;i < height;++ i) for (j = 0;j < width;++ j) ret[i][j] = nextInt();
			return ret;
		}

		public long[][] nextLong(int width, int height) {
			long[][] ret = new long[height][width];
			for (int i = 0, j;i < height;++ i) for (j = 0;j < width;++ j) ret[j][i] = nextLong();
			return ret;
		}

		public boolean[] nextBoolean(char T) {
			char[] s = next().toCharArray();
			boolean[] ret = new boolean[s.length];
			for (int i = 0;i < ret.length;++ i) ret[i] = s[i] == T;
			return ret;
		}

		public boolean[][] nextBoolean(char T, int height) {
			boolean[][] ret = new boolean[height][];
			for (int i = 0;i < ret.length;++ i) {
				char[] s = next().toCharArray();
				ret[i] = new boolean[s.length];
				for (int j = 0;j < ret[i].length;++ j) ret[i][j] = s[j] == T;
			}
			return ret;
		}

		public Point nextPoint() {
			return new Point(nextInt(), nextInt());
		}

		public Point[] nextPoint(int width) {
			Point[] ret = new Point[width];
			for (int i = 0;i < width;++ i) ret[i] = nextPoint();
			return ret;
		}

		@Override
		protected void finalize() throws Throwable {
			try {
				super.finalize();
			} finally {
				in.close();
				out.close();
				err.close();
			}
		}

		public boolean print(boolean b) {
			out.print(b);
			if (autoFlush) flush();
			return b;
		}

		public Object print(boolean b, Object t, Object f) {
			return b ? print(t) : print(f);
		}

		public char print(char c) {
			out.print(c);
			if (autoFlush) flush();
			return c;
		}

		public char[] print(char[] s) {
			out.print(s);
			return s;
		}

		public double print(double d) {
			out.print(d);
			if (autoFlush) flush();
			return d;
		}

		public double print(double d, int length) {
			if (d < 0) {
				out.print('-');
				d = -d;
			}
			d += Math.pow(10, -length) / 2;
			out.print((long)d);
			out.print('.');
			d -= (long)d;
			for (int i = 0;i < length;++ i) {
				d *= 10;
				out.print((int)d);
				d -= (int)d;
			}
			if (autoFlush) flush();
			return d;
		}

		public float print(float f) {
			out.print(f);
			if (autoFlush) flush();
			return f;
		}

		public int print(int i) {
			out.print(i);
			if (autoFlush) flush();
			return i;
		}

		public long print(long l) {
			out.print(l);
			if (autoFlush) flush();
			return l;
		}

		public Object print(Object obj) {
			if (obj.getClass().isArray()) {
				if (obj instanceof boolean[][]) print(obj, "\n", " ");
				else if (obj instanceof byte[][]) print(obj, "\n", " ");
				else if (obj instanceof short[][]) print(obj, "\n", " ");
				else if (obj instanceof int[][]) print(obj, "\n", " ");
				else if (obj instanceof long[][]) print(obj, "\n", " ");
				else if (obj instanceof float[][]) print(obj, "\n", " ");
				else if (obj instanceof double[][]) print(obj, "\n", " ");
				else if (obj instanceof char[][]) print(obj, "\n", " ");
				else if (obj instanceof Object[][]) print(obj, "\n", " ");
				else print(obj, " ");
			} else {
				out.print(obj);
				if (autoFlush) flush();
			}
			return obj;
		}

		public String print(String s) {
			out.print(s);
			if (autoFlush) flush();
			return s;
		}

		public Object print(Object array, String... parse) {
			print(array, 0, parse);
			if (autoFlush) flush();
			return array;
		}

		private Object print(Object array, int check, String... parse) {
			if (check >= parse.length) {
				if (array.getClass().isArray()) throw new IllegalArgumentException("not equal dimension");
				print(array);
				return array;
			}
			String str = parse[check];
			if (array instanceof Object[]) {
				Object[] obj = (Object[]) array;
				if (obj.length == 0) return array;
				print(obj[0], check + 1, parse);
				for (int i = 1;i < obj.length;++ i) {
					print(str);
					print(obj[i], check + 1, parse);
				}
				return array;
			}
			if (array instanceof Collection) {
				Iterator<?> iter = ((Collection<?>)array).iterator();
				if (!iter.hasNext()) return array;
				print(iter.next(), check + 1, parse);
				while(iter.hasNext()) {
					print(str);
					print(iter.next(), check + 1, parse);
				}
				return array;
			}
			if (!array.getClass().isArray()) throw new IllegalArgumentException("not equal dimension");
			if (check != parse.length - 1) throw new IllegalArgumentException("not equal dimension");
			if (array instanceof boolean[]) {
				boolean[] obj = (boolean[]) array;
				if (obj.length == 0) return array;
				print(obj[0]);
				for (int i = 1;i < obj.length;++ i) {
					print(str);
					print(obj[i]);
				}
			} else if (array instanceof byte[]) {
				byte[] obj = (byte[]) array;
				if (obj.length == 0) return array;
				print(obj[0]);
				for (int i = 1;i < obj.length;++ i) {
					print(str);
					print(obj[i]);
				}
				return array;
			} else if (array instanceof short[]) {
				short[] obj = (short[]) array;
				if (obj.length == 0) return array;
				print(obj[0]);
				for (int i = 1;i < obj.length;++ i) {
					print(str);
					print(obj[i]);
				}
			} else if (array instanceof int[]) {
				int[] obj = (int[]) array;
				if (obj.length == 0) return array;
				print(obj[0]);
				for (int i = 1;i < obj.length;++ i) {
					print(str);
					print(obj[i]);
				}
			} else if (array instanceof long[]) {
				long[] obj = (long[]) array;
				if (obj.length == 0) return array;
				print(obj[0]);
				for (int i = 1;i < obj.length;++ i) {
					print(str);
					print(obj[i]);
				}
			} else if (array instanceof float[]) {
				float[] obj = (float[]) array;
				if (obj.length == 0) return array;
				print(obj[0]);
				for (int i = 1;i < obj.length;++ i) {
					print(str);
					print(obj[i]);
				}
			} else if (array instanceof double[]) {
				double[] obj = (double[]) array;
				if (obj.length == 0) return array;
				print(obj[0]);
				for (int i = 1;i < obj.length;++ i) {
					print(str);
					print(obj[i]);
				}
			} else if (array instanceof char[]) {
				char[] obj = (char[]) array;
				if (obj.length == 0) return array;
				print(obj[0]);
				for (int i = 1;i < obj.length;++ i) {
					print(str);
					print(obj[i]);
				}
			} else throw new AssertionError();
			return array;
		}

		public Object[] print(String parse, Object... args) {
			print(args[0]);
			for (int i = 1;i < args.length;++ i) {
				print(parse);
				print(args[i]);
			}
			return args;
		}

		public Object[] printf(String format, Object... args) {
			out.printf(format, args);
			if (autoFlush) flush();
			return args;
		}

		public Object printf(Locale l, String format, Object... args) {
			out.printf(l, format, args);
			if (autoFlush) flush();
			return args;
		}

		public void println() {
			out.println();
			if (autoFlush) flush();
		}

		public boolean println(boolean b) {
			out.println(b);
			if (autoFlush) flush();
			return b;
		}

		public Object println(boolean b, Object t, Object f) {
			return b ? println(t) : println(f);
		}

		public char println(char c) {
			out.println(c);
			if (autoFlush) flush();
			return c;
		}

		public char[] println(char[] s) {
			out.println(s);
			if (autoFlush) flush();
			return s;
		}

		public double println(double d) {
			out.println(d);
			if (autoFlush) flush();
			return d;
		}

		public double println(double d, int length) {
			print(d, length);
			println();
			return d;
		}

		public float println(float f) {
			out.println(f);
			if (autoFlush) flush();
			return f;
		}

		public int println(int i) {
			out.println(i);
			if (autoFlush) flush();
			return i;
		}

		public long println(long l) {
			out.println(l);
			if (autoFlush) flush();
			return l;
		}

		public Object println(Object obj) {
			print(obj);
			println();
			return obj;
		}

		public String println(String s) {
			out.println(s);
			if (autoFlush) flush();
			return s;
		}

		public Object println(Object array, String... parse) {
			print(array, parse);
			println();
			return array;
		}

		public void flush() {
			out.flush();
			err.flush();
		}
	}

	public enum BoundType {
		CLOSED, OPEN;
	}

	public static class Range<C> implements Serializable{

		private static final long serialVersionUID = -4702828934863023392L;
		protected C lower;
		protected C upper;
		protected BoundType lowerType;
		protected BoundType upperType;
		private Comparator<? super C> comparator;

		protected Range(C lower, BoundType lowerType, C upper, BoundType upperType) {
			this(lower, lowerType, upper, upperType, null);
		}

		protected Range(C lower, BoundType lowerType, C upper, BoundType upperType, Comparator<? super C> comparator) {
			this.lower = lower;
			this.upper = upper;
			this.lowerType = lowerType;
			this.upperType = upperType;
			this.comparator = comparator;
		}

		public static <C extends Comparable<? super C>> Range<C> range(C lower, BoundType lowerType, C upper, BoundType upperType) {
			if (lower != null && upper != null) {
				int comp = lower.compareTo(upper);
				if (comp > 0) return new Range<C>(null, BoundType.CLOSED, null, BoundType.CLOSED);
				else if (comp == 0 && (lowerType == BoundType.OPEN || upperType == BoundType.OPEN))return new Range<C>(null, BoundType.CLOSED, null, BoundType.CLOSED);
			}
			return new Range<C>(lower, lowerType, upper, upperType);
		}

		public static <C> Range<C> range(C lower, BoundType lowerType, C upper, BoundType upperType, Comparator<? super C> comparator) {
			if (lower != null && upper != null) {
				int comp = comparator.compare(lower, upper);
				if (comp > 0) return new Range<C>(null, BoundType.CLOSED, null, BoundType.CLOSED, comparator);
				else if (comp == 0 && (lowerType == BoundType.OPEN || upperType == BoundType.OPEN)) return new Range<C>(null, BoundType.CLOSED, null, BoundType.CLOSED, comparator);
			}
			return new Range<C>(lower, lowerType, upper, upperType, comparator);
		}

		public static <C extends Comparable<? super C>> Range<C> all() {
			return range((C)null, BoundType.OPEN, null, BoundType.OPEN);
		}

		public static <C> Range<C> all(Comparator<? super C> comparator) {
			return range((C)null, BoundType.OPEN, null, BoundType.OPEN, comparator);
		}

		public static <C extends Comparable<? super C>> Range<C> atMost(C upper) {
			return range(null, BoundType.OPEN, upper, BoundType.CLOSED);
		}

		public static <C> Range<C> atMost(C upper, Comparator<? super C> comparator) {
			return range(null, BoundType.OPEN, upper, BoundType.CLOSED, comparator);
		}

		public static <C extends Comparable<? super C>> Range<C> lessThan(C upper) {
			return range(null, BoundType.OPEN, upper, BoundType.OPEN);
		}

		public static <C> Range<C> lessThan(C upper, Comparator<? super C> comparator) {
			return range(null, BoundType.OPEN, upper, BoundType.OPEN, comparator);
		}

		public static <C extends Comparable<? super C>> Range<C> downTo(C upper, BoundType boundType) {
			return range(null, BoundType.OPEN, upper, boundType);
		}

		public static <C> Range<C> downTo(C upper, BoundType boundType, Comparator<? super C> comparator) {
			return range(null, BoundType.OPEN, upper, boundType, comparator);
		}

		public static <C extends Comparable<? super C>> Range<C> atLeast(C lower) {
			return range(lower, BoundType.CLOSED, null, BoundType.OPEN);
		}

		public static <C> Range<C> atLeast(C lower, Comparator<? super C> comparator) {
			return range(lower, BoundType.CLOSED, null, BoundType.OPEN, comparator);
		}

		public static <C extends Comparable<? super C>> Range<C> greaterThan(C lower) {
			return range(lower, BoundType.OPEN, null, BoundType.OPEN);
		}

		public static <C> Range<C> greaterThan(C lower, Comparator<? super C> comparator) {
			return range(lower, BoundType.OPEN, null, BoundType.OPEN, comparator);
		}

		public static <C extends Comparable<? super C>> Range<C> upTo(C lower, BoundType boundType) {
			return range(lower, boundType, null, BoundType.OPEN);
		}

		public static <C> Range<C> upTo(C lower, BoundType boundType, Comparator<? super C> comparator) {
			return range(lower, boundType, null, BoundType.OPEN, comparator  );
		}

		public static <C extends Comparable<? super C>> Range<C> open(C lower, C upper) {
			return range(lower, BoundType.OPEN, upper, BoundType.OPEN);
		}

		public static <C> Range<C> open(C lower, C upper, Comparator<? super C> comparator) {
			return range(lower, BoundType.OPEN, upper, BoundType.OPEN, comparator);
		}

		public static <C extends Comparable<? super C>> Range<C> openClosed(C lower, C upper) {
			return range(lower, BoundType.OPEN, upper, BoundType.CLOSED);
		}

		public static <C> Range<C> openClosed(C lower, C upper, Comparator<? super C> comparator) {
			return range(lower, BoundType.OPEN, upper, BoundType.CLOSED, comparator);
		}

		public static <C extends Comparable<? super C>> Range<C> closedOpen(C lower, C upper) {
			return range(lower, BoundType.CLOSED, upper, BoundType.OPEN);
		}

		public static <C> Range<C> closedOpen(C lower, C upper, Comparator<? super C> comparator) {
			return range(lower, BoundType.CLOSED, upper, BoundType.OPEN, comparator);
		}

		public static <C extends Comparable<? super C>> Range<C> closed(C lower, C upper) {
			return range(lower, BoundType.CLOSED, upper, BoundType.CLOSED);
		}

		public static <C> Range<C> closed(C lower, C upper, Comparator<? super C> comparator) {
			return range(lower, BoundType.CLOSED, upper, BoundType.CLOSED, comparator);
		}

		public static <C extends Comparable<? super C>> Range<C> singleton(C value) {
			return range(value, BoundType.CLOSED, value, BoundType.CLOSED);
		}

		public static <C> Range<C> singleton(C value, Comparator<? super C> comparator) {
			return range(value, BoundType.CLOSED, value, BoundType.CLOSED, comparator);
		}

		public static <C extends Comparable<? super C>> Range<C> empty() {
			return range((C)null, BoundType.CLOSED, null, BoundType.CLOSED);
		}

		public static <C> Range<C> empty(Comparator<? super C> comparator) {
			return range((C)null, BoundType.CLOSED, null, BoundType.CLOSED, comparator);
		}

		public static <C extends Comparable<? super C>> Range<C> encloseAll(Iterable<C> values) {
			C lower = values.iterator().next();
			C upper = lower;
			for (C i : values) {
				if (lower.compareTo(i) > 0) lower = i;
				if (upper.compareTo(i) < 0) upper = i;
			}
			return range(lower, BoundType.CLOSED, upper, BoundType.CLOSED);
		}

		public static <C> Range<C> encloseAll(Iterable<C> values, Comparator<? super C> comparator) {
			C lower = values.iterator().next();
			C upper = lower;
			for (C i : values) {
				if (comparator.compare(lower, i) > 0) lower = i;
				if (comparator.compare(upper, i) < 0) upper = i;
			}
			return range(lower, BoundType.CLOSED, upper, BoundType.CLOSED, comparator);
		}

		protected int compareLower(C value) {
			return compareLower(value, BoundType.CLOSED);
		}

		protected int compareLower(C value, BoundType boundType) {
			return compareLower(lower, lowerType, value, boundType);
		}

		protected int compareLower(C lower, BoundType lowerType, C value) {
			return compareLower(lower, lowerType, value, BoundType.CLOSED);
		}

		protected int compareLower(C lower, BoundType lowerType, C value, BoundType boundType) {
			if (lower == null) return value == null ? 0 : -1;
			else if (value == null) return 1;
			int compare;
			if (comparator == null) {
				@SuppressWarnings("unchecked")
				Comparable<C> comp = (Comparable<C>)lower;
				compare = comp.compareTo(value);
			} else compare = comparator.compare(lower, value);
			if (compare == 0) {
				if (lowerType == BoundType.CLOSED) -- compare;
				if (boundType == BoundType.CLOSED) ++ compare;
			}
			return compare;
		}

		protected int compareUpper(C value) {
			return compareUpper(value, BoundType.CLOSED);
		}

		protected int compareUpper(C value, BoundType boundType) {
			return compareUpper(upper, upperType, value, boundType);
		}

		protected int compareUpper(C upper, BoundType upperType, C value) {
			return compareUpper(upper, upperType, value, BoundType.CLOSED);
		}

		protected int compareUpper(C upper, BoundType upperType, C value, BoundType boundType) {
			if (upper == null) return value == null ? 0 : 1;
			if (value == null) return -1;
			int compare;
			if (comparator == null) {
				@SuppressWarnings("unchecked")
				Comparable<C> comp = (Comparable<C>)upper;
				compare = comp.compareTo(value);
			} else compare = comparator.compare(upper, value);
			if (compare == 0) {
				if (upperType == BoundType.CLOSED) ++ compare;
				if (boundType == BoundType.CLOSED) -- compare;
			}
			return compare;
		}

		public boolean hasLowerBound() {
			return lower != null;
		}

		public C lowerEndpoint() {
			if (hasLowerBound()) return lower;
			throw new IllegalStateException();
		}

		public BoundType lowerBoundType() {
			if (hasLowerBound()) return lowerType;
			throw new IllegalStateException();
		}

		public boolean hasUpperBound() {
			return upper != null;
		}

		public C upperEndpoint() {
			if (hasUpperBound()) return upper;
			throw new IllegalStateException();
		}

		public BoundType upperBoundType() {
			if (hasUpperBound()) return upperType;
			throw new IllegalStateException();
		}

		/**
		 * この区間が空集合か判定します。
		 * @return 空集合ならばtrue
		 */
		public boolean isEmpty() {
			return lower == null && upper == null && lowerType == BoundType.CLOSED;
		}

		/**
		 * 与えられた引数が区間の左側に位置するか判定します。<br>
		 * 接する場合は区間の左側ではないと判定します。
		 * @param value 調べる引数
		 * @return 区間の左側に位置するならtrue
		 */
		public boolean isLess(C value) {
			return isLess(value, BoundType.CLOSED);
		}

		protected boolean isLess(C value, BoundType boundType) {
			return compareLower(value, boundType) > 0;
		}

		/**
		 * 与えられた引数が区間の右側に位置するか判定します。<br>
		 * 接する場合は区間の右側ではないと判定します。
		 * @param value 調べる引数
		 * @return 区間の右側に位置するならtrue
		 */
		public boolean isGreater(C value) {
			return isGreater(value, BoundType.CLOSED);
		}

		private boolean isGreater(C value, BoundType boundType) {
			return compareUpper(value, boundType) < 0;
		}

		/**
		 * 与えられた引数が区間内に位置するか判定します。<br>
		 * 接する場合も区間内に位置すると判定します。
		 * @param value 調べる引数
		 * @return 区間内に位置するならtrue
		 */
		public boolean contains(C value) {
			return !isLess(value) && !isGreater(value) && !isEmpty();
		}

		/**
		 * 与えられた引数すべてが区間内に位置するか判定します。<br>
		 * 接する場合も区間内に位置すると判定します。
		 * @param value 調べる要素
		 * @return 全ての要素が区間内に位置するならtrue
		 */
		public boolean containsAll(Iterable<? extends C> values) {
			for (C i : values) if (!contains(i)) return false;
			return true;
		}

		/**
		 * 与えられた区間がこの区間に内包されるか判定します。<br>
		 *
		 * @param other
		 * @return 与えられた区間がこの区間に内包されるならtrue
		 */
		public boolean encloses(Range<C> other) {
			return !isLess(other.lower, other.lowerType) && !isGreater(other.upper, other.upperType);
		}

		/**
		 * 与えられた区間がこの区間と公差するか判定します。<br>
		 * 接する場合は公差するものとします。
		 * @param value 調べる引数
		 * @return 区間が交差するならtrue
		 */
		public boolean isConnected(Range<C> other) {
			if (this.isEmpty() || other.isEmpty()) return false;
			C lower, upper;
			BoundType lowerType, upperType;
			if (isLess(other.lower, other.lowerType)) {
				lower = other.lower;
				lowerType = other.lowerType;
			} else {
				lower = this.lower;
				lowerType = this.lowerType;
			}
			if (isGreater(other.upper, other.upperType)) {
				upper = other.upper;
				upperType = other.upperType;
			} else {
				upper = this.upper;
				upperType = this.upperType;
			}
			if (lower == null || upper == null) return true;
			int comp = compareLower(lower, lowerType, upper, upperType);
			return comp <= 0;
		}
		/**
		 * この区間との積集合を返します。
		 * @param connectedRange 積集合を求める区間
		 * @return 積集合
		 */
		public Range<C> intersection(Range<C> connectedRange) {
			if (this.isEmpty() || connectedRange.isEmpty()) {
				if (comparator == null) return new Range<C>(null, BoundType.CLOSED, null, BoundType.CLOSED);
				return empty(comparator);
			}
			C lower, upper;
			BoundType lowerType, upperType;
			if (isLess(connectedRange.lower, connectedRange.lowerType)) {
				lower = connectedRange.lower;
				lowerType = connectedRange.lowerType;
			} else {
				lower = this.lower;
				lowerType = this.lowerType;
			}
			if (isGreater(connectedRange.upper, connectedRange.upperType)) {
				upper = connectedRange.upper;
				upperType = connectedRange.upperType;
			} else {
				upper = this.upper;
				upperType = this.upperType;
			}
			if (comparator == null) {
				return new Range<C>(lower, lowerType, upper, upperType);
			}
			return range(lower, lowerType, upper, upperType, comparator);
		}

		/**
		 * この区間との和集合を返します。
		 * @param other 和集合を求める区間
		 * @return 和集合
		 */
		public Range<C> span(Range<C> other) {
			if (other.isEmpty()) return new Range<C>(lower, lowerType, upper, upperType);
			C lower, upper;
			BoundType lowerType, upperType;
			if (isLess(other.lower, other.lowerType)) {
				lower = this.lower;
				lowerType = this.lowerType;
			} else {
				lower = other.lower;
				lowerType = other.lowerType;
			}
			if (isGreater(other.upper, other.upperType)) {
				upper = this.upper;
				upperType = this.upperType;
			} else {
				upper = other.upper;
				upperType = other.upperType;
			}
			return new Range<C>(lower, lowerType, upper, upperType, comparator);
		}

		@Override
		public boolean equals(Object object) {
			if (this == object) return true;
			if (object instanceof Range) {
				@SuppressWarnings("unchecked")
				Range<C> comp = (Range<C>) object;
				return compareLower(comp.lower, comp.lowerType) == 0 && compareUpper(comp.upper, comp.upperType) == 0 && lowerType == comp.lowerType && upperType == comp.upperType;
			}
			return false;
		}

		@Override
		public int hashCode() {
			if (lower == null && upper == null) return 0;
			else if (lower == null) return upper.hashCode();
			else if (upper == null) return lower.hashCode();
			return lower.hashCode() ^ upper.hashCode();
		}

		@Override
		public String toString() {
			if (isEmpty()) return "()";
			return (lowerType == BoundType.OPEN ? "(" : "[") + (lower == null ? "" : lower.toString()) + ".." + (upper == null ? "" : upper.toString()) + (upperType == BoundType.OPEN ? ")" : "]");
		}
	}

	public static class IterableRange<C> extends Range<C> implements Iterable<C>{

		private static final long serialVersionUID = 9065915259748260688L;
		protected UnaryOperator<C> func;

		protected IterableRange(C lower, BoundType lowerType, C upper, BoundType upperType, UnaryOperator<C> func) {
			super(lower, lowerType, upper, upperType);
			this.func = func;
		}

		public static <C extends Comparable<? super C>> IterableRange<C> range(C lower, BoundType lowerType, C upper, BoundType upperType, UnaryOperator<C> func) {
			if (lower == null || upper == null) return new IterableRange<C>(null, BoundType.CLOSED, null, BoundType.CLOSED, func);
			int comp = lower.compareTo(upper);
			if (comp > 0) return new IterableRange<C>(null, BoundType.CLOSED, null, BoundType.CLOSED, func);
			else if (comp == 0 && (lowerType == BoundType.OPEN || upperType == BoundType.OPEN)) return new IterableRange<C>(null, BoundType.CLOSED, null, BoundType.CLOSED, func);
			return new IterableRange<C>(lower, lowerType, upper, upperType, func);
		}

		public static <C extends Comparable<? super C>> IterableRange<C> open(C lower, C upper, UnaryOperator<C> func) {
			if (lower == null) return new IterableRange<C>(null, BoundType.CLOSED, null, BoundType.CLOSED, func);
			return range(func.apply(lower), BoundType.CLOSED, upper, BoundType.OPEN, func);
		}

		public static <C extends Comparable<? super C>> IterableRange<C> openClosed(C lower, C upper, UnaryOperator<C> func) {
			if (lower == null) return new IterableRange<C>(null, BoundType.CLOSED, null, BoundType.CLOSED, func);
			return range(func.apply(lower), BoundType.CLOSED, upper, BoundType.CLOSED, func);
		}

		public static <C extends Comparable<? super C>> IterableRange<C> closedOpen(C lower, C upper, UnaryOperator<C> func) {
			return range(lower, BoundType.CLOSED, upper, BoundType.OPEN, func);
		}

		public static <C extends Comparable<? super C>> IterableRange<C> closed(C lower, C upper, UnaryOperator<C> func) {
			return range(lower, BoundType.CLOSED, upper, BoundType.CLOSED, func);
		}

		public static <C extends Comparable<? super C>> IterableRange<C> singleton(C value, UnaryOperator<C> func) {
			return range(value, BoundType.CLOSED, value, BoundType.CLOSED, func);
		}

		protected class Iter implements Iterator<C> {
			C now;
			Iter() {
				now = lower;
			}
			@Override
			public final boolean hasNext() {
				return !isGreater(now);
			}

			@Override
			public final C next() {
				C ret = now;
				now = func.apply(now);
				return ret;
			}

			@Override
			public final void remove() {
				throw new UnsupportedOperationException();
			}
		}

		protected class EmptyIter implements Iterator<C> {

			@Override
			public boolean hasNext() {
				return false;
			}

			@Override
			public C next() {
				return null;
			}

			@Override
			public final void remove() {
				throw new UnsupportedOperationException();
			}

		}

		@Override
		public Iterator<C> iterator() {
			return lower == null || upper == null ? new EmptyIter() : new Iter();
		}

		public int getDistance() {
			C check = upper;
			int ret = 0;
			while (lower != check) {
				check = func.apply(check);
				++ ret;
			}
			return ret;
		}
	}

	public static class IntRange extends IterableRange<Integer>{

		private static final long serialVersionUID = 5623995336491967216L;
		private final boolean useFastIter;

		private static class Next implements UnaryOperator<Integer> {

			@Override
			public Integer apply(Integer value) {
				return value + 1;
			}
		}

		protected IntRange() {
			super(null, BoundType.CLOSED, null, BoundType.CLOSED, new Next());
			useFastIter = true;
		}

		protected IntRange(UnaryOperator<Integer> func) {
			super(null, BoundType.CLOSED, null, BoundType.CLOSED, func);
			useFastIter = false;
		}

		protected IntRange(int lower, BoundType lowerType, int upper, BoundType upperType) {
			super(lower, lowerType, upper, upperType, new Next());
			useFastIter = true;
		}

		protected IntRange(int lower, BoundType lowerType, int upper, BoundType upperType, UnaryOperator<Integer> func) {
			super(lower, lowerType, upper, upperType, func);
			useFastIter = false;
		}

		public static IntRange range(int lower, BoundType lowerType, int upper, BoundType upperType) {
			if (lower > upper) return new IntRange();
			if (lowerType == BoundType.OPEN) ++ lower;
			if (upperType == BoundType.OPEN) -- upper;
			return new IntRange(lower, BoundType.CLOSED, upper, BoundType.CLOSED);
		}

		public static IntRange range(int lower, BoundType lowerType, int upper, BoundType upperType, UnaryOperator<Integer> func) {
			if (lower > upper) return new IntRange(func);
			if (lowerType == BoundType.OPEN) ++ lower;
			if (upperType == BoundType.OPEN) -- upper;
			return new IntRange(lower, BoundType.CLOSED, upper, BoundType.CLOSED, func);
		}

		public static IntRange open(int lower, int upper) {
			return range(lower, BoundType.OPEN, upper, BoundType.OPEN);
		}

		public static IntRange open(int lower, int upper, UnaryOperator<Integer> func) {
			return range(lower, BoundType.OPEN, upper, BoundType.OPEN, func);
		}

		public static IntRange open(int upper) {
			return range(0, BoundType.CLOSED, upper, BoundType.OPEN);
		}

		public static IntRange open(int upper, UnaryOperator<Integer> func) {
			return range(0, BoundType.CLOSED, upper, BoundType.OPEN, func);
		}

		public static IntRange openClosed(int lower, int upper) {
			return range(lower, BoundType.OPEN, upper, BoundType.CLOSED);
		}

		public static IntRange openClosed(int lower, int upper, UnaryOperator<Integer> func) {
			return range(lower, BoundType.OPEN, upper, BoundType.CLOSED, func);
		}

		public static IntRange closedOpen(int lower, int upper) {
			return range(lower, BoundType.CLOSED, upper, BoundType.OPEN);
		}

		public static IntRange closedOpen(int lower, int upper, UnaryOperator<Integer> func) {
			return range(lower, BoundType.CLOSED, upper, BoundType.OPEN, func);
		}

		public static IntRange closed(int lower, int upper) {
			return range(lower, BoundType.CLOSED, upper, BoundType.CLOSED);
		}

		public static IntRange closed(int lower, int upper, UnaryOperator<Integer> func) {
			return range(lower, BoundType.CLOSED, upper, BoundType.CLOSED, func);
		}

		public static IntRange closed(int upper) {
			return range(0, BoundType.CLOSED, upper, BoundType.CLOSED);
		}

		public static IntRange closed(int upper, UnaryOperator<Integer> func) {
			return range(0, BoundType.CLOSED, upper, BoundType.CLOSED, func);
		}

		public static IntRange singleton(int value) {
			return range(value, BoundType.CLOSED, value, BoundType.CLOSED);
		}

		public static IntRange singleton(int value, UnaryOperator<Integer> func) {
			return range(value, BoundType.CLOSED, value, BoundType.CLOSED, func);
		}

		private class FastIter implements Iterator<Integer> {
			int now;
			public FastIter() {
				now = lower;
			}
			@Override
			public final boolean hasNext() {
				return now <= upper;
			}

			@Override
			public final Integer next() {
				return now++;
			}

			@Override
			public final void remove() {
				throw new UnsupportedOperationException();
			}
		}

		private class Iter implements Iterator<Integer> {
			int now;
			public Iter() {
				now = lower;
			}
			@Override
			public final boolean hasNext() {
				return now <= upper;
			}

			@Override
			public final Integer next() {
				int ret = now;
				now = func.apply(now);
				return ret;
			}

			@Override
			public final void remove() {
				throw new UnsupportedOperationException();
			}
		}

		@Override
		public Iterator<Integer> iterator() {
			return lower == null || upper == null ? new EmptyIter() : useFastIter ? new FastIter() : new Iter();
		}

		@Override
		public int getDistance() {
			int ret = upper - lower;
			if (upperType == BoundType.CLOSED) ++ ret;
			return ret;
		}

		public int getClosedLower() {
			return lower;
		}

		public int getOpenLower() {
			return lower - 1;
		}

		public int getClosedUpper() {
			return upperType == BoundType.CLOSED ? upper : upper - 1;
		}

		public int getOpenUpper() {
			return upperType == BoundType.CLOSED ? upper + 1 : upper;
		}
	}

	/**
	 * 演算が結合法則を満たすことを示すために使用するマーカー・インターフェースです。
	 * @author 31536000
	 *
	 * @param <T> 二項演算の型
	 */
	public interface Associative<T> extends BinaryOperator<T>{
		/**
		 * repeat個のelementを順次演算した値を返します。
		 * @param element 演算する値
		 * @param repeat 繰り返す回数、1以上であること
		 * @return 演算を+として、element + element + ... + elementと演算をrepeat-1回行った値
		 */
		public default T hyper(T element, int repeat) {
			if (repeat < 1) throw new IllegalArgumentException("undefined operation");
			T ret = element;
			-- repeat;
			for (T mul = element;repeat > 0;repeat >>= 1, mul = apply(mul, mul)) if ((repeat & 1) != 0) ret = apply(ret, mul);
			return ret;
		}
	}

	/**
	* この演算が逆元を持つことを示すために使用するマーカー・インターフェースです。
	 * @author 31536000
	 *
	 * @param <T> 二項演算の型
	 */
	public interface Inverse<T> extends BinaryOperator<T>{
		public T inverse(T element);
	}

	/**
	 * 演算が交換法則を満たすことを示すために使用するマーカー・インターフェースです。
	 * @author 31536000
	 *
	 * @param <T> 二項演算の型
	 */
	public interface Commutative<T> extends BinaryOperator<T>{

	}

	/**
	 * 演算が単位元を持つことを示すために使用するマーカー・インターフェースです。
	 * @author 31536000
	 *
	 * @param <T> 二項演算の型
	 */
	public interface Unit<T> extends BinaryOperator<T>{
		/**
		 * 単位元を返します。
		 * @return 単位元
		 */
		public T unit();
	}

	/**
	 * 演算が群であることを示すために使用するマーカー・インターフェースです。
	 * @author 31536000
	 *
	 * @param <T> 二項演算の型
	 */
	public interface Group<T> extends Monoid<T>, Inverse<T>{
		/**
		 * repeat個のelementを順次演算した値を返します。
		 * @param element 演算する値
		 * @param repeat 繰り返す回数
		 * @return 演算を+として、element + element + ... + elementと演算をrepeat-1回行った値
		 */
		@Override
		public default T hyper(T element, int repeat) {
			T ret = unit();
			if (repeat < 0) {
				repeat = -repeat;
				for (T mul = element;repeat > 0;repeat >>= 1, mul = apply(mul, mul)) if ((repeat & 1) != 0) ret = apply(ret, mul);
				return inverse(ret);
			}
			for (T mul = element;repeat > 0;repeat >>= 1, mul = apply(mul, mul)) if ((repeat & 1) != 0) ret = apply(ret, mul);
			return ret;
		}
	}

	/**
	 * 演算がモノイドであることを示すために使用するマーカー・インターフェースです。
	 * @author 31536000
	 *
	 * @param <T> 二項演算の型
	 */
	public interface Monoid<T> extends Associative<T>, Unit<T> {
		/**
		 * repeat個のelementを順次演算した値を返します。
		 * @param element 演算する値
		 * @param repeat 繰り返す回数、0以上であること
		 * @return 演算を+として、element + element + ... + elementと演算をrepeat-1回行った値
		 */
		@Override
		public default T hyper(T element, int repeat) {
			if (repeat < 0) throw new IllegalArgumentException("undefined operation");
			T ret = unit();
			for (T mul = element;repeat > 0;repeat >>= 1, mul = apply(mul, mul)) if ((repeat & 1) != 0) ret = apply(ret, mul);
			return ret;
		}
	}

	/**
	 * 演算が可換モノイドであることを示すために使用するマーカー・インターフェースです。
	 * @author 31536000
	 *
	 * @param <T> 二項演算の型
	 */
	public interface CommutativeMonoid<T> extends Monoid<T>, Commutative<T> {

	}

	/**
	 * 演算がアーベル群(可換群)であることを示すために使用するマーカー・インターフェースです。
	 * @author 31536000
	 *
	 * @param <T> 二項演算の型
	 */
	public interface Abelian<T> extends Group<T>, CommutativeMonoid<T> {

	}

	/**
	 * 演算が半環であることを示すために使用するマーカー・インターフェースです。
	 * @author 31536000
	 *
	 * @param <T> 二項演算の型
	 * @param <A> 和に関する演算
	 * @param <M> 積に関する演算
	 */
	public interface Semiring<T, A extends CommutativeMonoid<T>, M extends Monoid<T>> {
		public A getAddition();
		public M getMultiplication();
	}

	/**
	 * 演算が環であることを示すために使用するマーカー・インターフェースです。
	 * @author 31536000
	 *
	 * @param <T> 二項演算の型
	 * @param <A> 和に関する演算
	 * @param <M> 積に関する演算
	 */
	public interface Ring<T, A extends Abelian<T>, M extends Monoid<T>> extends Semiring<T, A, M>{

	}

	/**
	 * 演算が可換環に属することを示すために使用するマーカー・インターフェースです。
	 * @author 31536000
	 *
	 * @param <T> 二項演算の型
	 * @param <A> 和に関する演算
	 * @param <M> 積に関する演算
	 */
	public interface CommutativeRing<T, A extends Abelian<T>, M extends CommutativeMonoid<T>> extends Ring<T, A, M>{

	}

	public static class ModInteger extends Number implements CommutativeRing<ModInteger, Abelian<ModInteger>, CommutativeMonoid<ModInteger>>{

		private static final long serialVersionUID = -8595710127161317579L;
		private final int mod;
		private int num;

		private final Addition add;
		private final Multiplication mul;

		private class Addition implements Abelian<ModInteger> {

			@Override
			public ModInteger unit() {
				return new ModInteger(mod, 0);
			}

			@Override
			public ModInteger inverse(ModInteger element) {
				return new ModInteger(element, element.mod - element.num);
			}

			@Override
			public ModInteger apply(ModInteger left, ModInteger right) {
				return new ModInteger(left).addEqual(right);
			}
		}

		private class Multiplication implements Abelian<ModInteger> {

			@Override
			public ModInteger unit() {
				return new ModInteger(mod, 1);
			}

			@Override
			public ModInteger apply(ModInteger left, ModInteger right) {
				return new ModInteger(left).multiplyEqual(right);
			}

			@Override
			public ModInteger inverse(ModInteger element) {
				return new ModInteger(element, element.inverse(element.num));
			}

		}

		public ModInteger(int mod) {
			this.mod = mod;
			num = 0;
			add = new Addition();
			mul = new Multiplication();
		}

		public ModInteger(int mod, int num) {
			this.mod = mod;
			this.num = validNum(num);
			add = new Addition();
			mul = new Multiplication();
		}

		public ModInteger(ModInteger n) {
			mod = n.mod;
			num = n.num;
			add = n.add;
			mul = n.mul;
		}

		private ModInteger(ModInteger n, int num) {
			mod = n.mod;
			this.num = num;
			add = n.add;
			mul = n.mul;
		}

		private int validNum(int n) {
			n %= mod;
			if (n < 0) n += mod;
			return n;
		}

		private int validNum(long n) {
			n %= mod;
			if (n < 0) n += mod;
			return (int)n;
		}

		protected int inverse(int n) {
			int m = mod, u = 0, v = 1, t;
			while(n != 0) {
				t = m / n;
				m -= t * n;
				u -= t * v;
				if (m != 0) {
					t = n / m;
					n -= t * m;
					v -= t * u;
				} else {
					v %= mod;
					if (v < 0) v += mod;
					return v;
				}
			}
			u %= mod;
			if (u < 0) u += mod;
			return u;
		}

		public boolean isPrime(int n) {
			if ((n & 1) == 0) return false; // 偶数
			for (int i = 3, j = 8, k = 9;k <= n;i += 2, k += j += 8) if (n % i == 0) return false;
			return true;
		}

		@Override
		public int intValue() {
			return num;
		}

		@Override
		public long longValue() {
			return num;
		}

		@Override
		public float floatValue() {
			return num;
		}

		@Override
		public double doubleValue() {
			return num;
		}

		public ModInteger add(int n) {
			return new ModInteger(this).addEqual(n);
		}

		public ModInteger add(long n) {
			return new ModInteger(this).addEqual(n);
		}

		public ModInteger add(ModInteger n) {
			return new ModInteger(this).addEqual(n);
		}

		public ModInteger addEqual(int n) {
			num = validNum(num + n);
			return this;
		}

		public ModInteger addEqual(long n) {
			num = validNum(num + n);
			return this;
		}

		public ModInteger addEqual(ModInteger n) {
			if ((num += n.num) >= mod) num -= mod;
			return this;
		}

		public ModInteger subtract(int n) {
			return new ModInteger(this).subtractEqual(n);
		}

		public ModInteger subtract(long n) {
			return new ModInteger(this).subtractEqual(n);
		}

		public ModInteger subtract(ModInteger n) {
			return new ModInteger(this).subtractEqual(n);
		}

		public ModInteger subtractEqual(int n) {
			num = validNum(num - n);
			return this;
		}

		public ModInteger subtractEqual(long n) {
			num = validNum(num - n);
			return this;
		}

		public ModInteger subtractEqual(ModInteger n) {
			if ((num -= n.num) < 0) num += mod;
			return this;
		}

		public ModInteger multiply(int n) {
			return new ModInteger(this).multiplyEqual(n);
		}

		public ModInteger multiply(long n) {
			return new ModInteger(this).multiplyEqual(n);
		}

		public ModInteger multiply(ModInteger n) {
			return new ModInteger(this).multiplyEqual(n);
		}

		public ModInteger multiplyEqual(int n) {
			num = (int)((long)num * n % mod);
			if (num < 0) num += mod;
			return this;
		}

		public ModInteger multiplyEqual(long n) {
			return multiplyEqual((int) (n % mod));
		}

		public ModInteger multiplyEqual(ModInteger n) {
			num = (int)((long)num * n.num % mod);
			return this;
		}

		public ModInteger divide(int n) {
			return new ModInteger(this).divideEqual(n);
		}

		public ModInteger divide(long n) {
			return new ModInteger(this).divideEqual(n);
		}

		public ModInteger divide(ModInteger n) {
			return new ModInteger(this).divideEqual(n);
		}

		public ModInteger divideEqual(int n) {
			num = (int)((long)num * inverse(validNum(n)) % mod);
			return this;
		}

		public ModInteger divideEqual(long n) {
			return divideEqual((int)(n % mod));
		}

		public ModInteger divideEqual(ModInteger n) {
			num = (int)((long)num * n.inverse(n.num) % mod);
			return this;
		}

		public ModInteger pow(int n) {
			return new ModInteger(this).powEqual(n);
		}

		public ModInteger pow(long n) {
			return new ModInteger(this).powEqual(n);
		}

		public ModInteger pow(ModInteger n) {
			return new ModInteger(this).powEqual(n);
		}

		public ModInteger powEqual(int n) {
			long ans = 1, num = this.num;
			if (n < 0) {
				n = -n;
				while (n != 0) {
					if ((n & 1) != 0) ans = ans * num % mod;
					n >>>= 1;
			num = num * num % mod;
				}
				this.num = inverse((int)ans);
				return this;
			}
			while (n != 0) {
				if ((n & 1) != 0) ans = ans * num % mod;
				n >>>= 1;
					num = num * num % mod;
			}
			this.num = (int)ans;
			return this;
		}
		public ModInteger powEqual(long n) {
			return powEqual((int)(n % (mod - 1)));
		}

		public ModInteger powEqual(ModInteger n) {
			long num = this.num;
			this.num = 1;
			int mul = n.num;
			while (mul != 0) {
				if ((mul & 1) != 0) this.num *= num;
				mul >>>= 1;
				num *= num;
				num %= mod;
			}
			return this;
		}

		public ModInteger equal(int n) {
			num = validNum(n);
			return this;
		}

		public ModInteger equal(long n) {
			num = validNum(n);
			return this;
		}

		public ModInteger equal(ModInteger n) {
			num = n.num;
			return this;
		}

		public int toInt() {
			return num;
		}

		public int getMod() {
			return mod;
		}

		@Override
		public boolean equals(Object x) {
			if (x instanceof ModInteger) return ((ModInteger)x).num == num && ((ModInteger)x).mod == mod;
			return false;
		}

		@Override
		public int hashCode() {
			return num ^ mod;
		}

		@Override
		public String toString() {
			return String.valueOf(num);
		}

		@Deprecated
		public String debug() {
			int min = num, ans = 1;
			for (int i = 2;i < min;++ i) {
				int tmp = multiply(i).num;
				if (min > tmp) {
					min = tmp;
					ans = i;
				}
			}
			return min + "/" + ans;
		}

		@Override
		public Addition getAddition() {
			return add;
		}

		@Override
		public Multiplication getMultiplication() {
			return mul;
		}
	}

	/**
	 * 素数を法とする演算上で、組み合わせの計算を高速に行います。
	 * @author 31536000
	 *
	 */
	public static class ModUtility {
		private final int mod, totient;
		private int[] fact, inv, invfact;
		/**
		 * modを法とする
		 * @param mod
		 */
		public ModUtility(int mod) {
			this(mod, 2);
		}

		public ModUtility(int mod, int calc) {
			if (mod <= 0) throw new IllegalArgumentException("illegal mod: " + mod);
			this.mod = mod;
			int totient = mod;
			for (int i = 2;i * i <= mod;++ i) {
				if (mod % i == 0) {
					totient = totient / i * (i - 1);
					while ((mod %= i) % i == 0);
				}
			}
			this.totient = totient;
			precalc(calc);
		}

		public void precalc(int calc) {
			if (calc < 2) calc = 2;
			fact = new int[calc];
			inv = new int[calc];
			invfact = new int[calc];
			fact[0] = invfact[0] = fact[1] = invfact[1] = inv[1] = 1;
			for (int i = 2;i < calc;++ i) {
				fact[i] = (int)((long)fact[i - 1] * i % mod);
				inv[i] = (int)(mod - (long)inv[mod % i] * (mod / i) % mod);
				invfact[i] = (int)((long)invfact[i - 1] * inv[i] % mod);
			}
		}

		public ModInteger create() {
			return create(0);
		}

		public ModInteger create(int n) {
			return new ModInt(n);
		}

		private class ModInt extends ModInteger {

			private static final long serialVersionUID = -2435281861935422575L;

			public ModInt(int n) {
				super(mod, n);
			}

			@Override
			protected int inverse(int n) {
				return ModUtility.this.inverse(n);
			}
		}

		public int inverse(int n) {
			try {
				if (inv.length > n) return inv[n];
				int m = mod, u = 0, v = 1, t;
				while(n != 0) {
					t = m / n;
					m -= t * n;
					u -= t * v;
					if (m != 0) {
						t = n / m;
						n -= t * m;
						v -= t * u;
					} else {
						v %= mod;
						if (v < 0) v += mod;
						return v;
					}
				}
				u %= mod;
				if (u < 0) u += mod;
				return u;
			} catch (ArrayIndexOutOfBoundsException e) {
				throw new IllegalArgumentException();
			}
		}

		public int factorial(int n) {
			try {
				if (fact.length > n) return fact[n];
				long ret = fact[fact.length - 1];
				for (int i = fact.length;i <= n;++ i) ret = ret * i % mod;
				return (int)ret;
			} catch (ArrayIndexOutOfBoundsException e) {
				throw new IllegalArgumentException();
			}
		}

		public int permutation(int n, int k) {
			if (k < 0) throw new IllegalArgumentException();
			if (n < k) return 0;
			if (fact.length > n) return (int)((long)fact[n] * invfact[n - k] % mod);
			long ret = 1;
			for (int i = n - k + 1;i <= n;++ i) ret = ret * i % mod;
			return (int)ret;
		}

		public int combination(int n, int k) {
			if (k < 0) throw new IllegalArgumentException();
			if (n < k) return 0;
			if (fact.length > n) return (int)((long)fact[n] * invfact[k] % mod * invfact[n - k] % mod);
			long ret = 1;
			if (n < 2 * k) k = n - k;
			if (invfact.length > k) ret = invfact[k];
			else ret = inverse(factorial(k));
			for (int i = n - k + 1;i <= n;++ i) ret = ret * i % mod;
			return (int)ret;
		}

		public int multinomial(int n, int... k) {
			int sum = 0;
			for (int i : k) sum += i;
			long ret = factorial(n);
			if (fact.length > n) {
				for (int i : k) {
					if (i < 0) throw new IllegalArgumentException();
					ret = ret * invfact[i] % mod;
					sum += i;
				}
				if (sum > n) return 0;
				ret = ret * invfact[n - sum] % mod;
			} else {
				for (int i : k) {
					if (i < 0) throw new IllegalArgumentException();
					if (invfact.length > i) ret = ret * invfact[i] % mod;
					else ret = ret * inverse(factorial(i)) % mod;
					sum += i;
				}
				if (sum > n) return 0;
				if (invfact.length > n - sum) ret = ret * invfact[n - sum] % mod;
				else ret = ret * inverse(factorial(n - sum)) % mod;
			}
			return (int)ret;
		}

		public int multichoose(int n, int k) {
			return combination(mod(n + k - 1), k);
		}

		public int catalan(int n) {
			return divide(combination(mod(2 * n), n), mod(n + 1));
		}

		public int pow(int n, int m) {
			long ans = 1, num = n;
			if (m < 0) {
				m = -m;
				while (m != 0) {
					if ((m & 1) != 0) ans = ans * num % mod;
					m >>>= 1;
			num = num * num % mod;
				}
				return inverse((int)ans);
			}
			while (m != 0) {
				if ((m & 1) != 0) ans = ans * num % mod;
				m >>>= 1;
			num = num * num % mod;
			}
			return (int)ans;
		}

		public int pow(long n, long m) {
			return pow((int)(n % mod), (int)(m % (mod - 1)));
		}

		public int totient() {
			return totient;
		}

		public boolean isPrime() {
			return totient == mod - 1;
		}

		public int mod(int n) {
			return (n %= mod) < 0 ? n + mod : n;
		}

		public int mod(long n) {
			return (int)((n %= mod) < 0 ? n + mod : n);
		}

		public int add(int n, int m) {
			return mod(n + m);
		}

		public int add(long n, long m) {
			return mod(n + m);
		}

		public int subtract(int n, int m) {
			return mod(n - m);
		}

		public int subtract(long n, long m) {
			return mod(n - m);
		}

		public int multiply(int n, int m) {
			int ans = (int)((long)n * m % mod);
			return ans < 0 ? ans + mod : ans;
		}

		public int multiply(long n, long m) {
			return multiply(mod(n), mod(m));
		}

		public int divide(int n, int m) {
			return multiply(n, inverse(mod(m)));
		}

		public int divide(long n, long m) {
			return multiply(n, inverse(mod(m)));
		}

		public ModInteger lagrangePolynomial(ModInteger[] f, int x) {
			if (f.length > x) return f[x];
			if (x > fact.length) precalc(x);
			ModInteger ret = create(0);
			ModInteger[] dp = new ModInteger[f.length], dp2 = new ModInteger[f.length];
			dp[0] = create(1);
			dp2[f.length - 1] = create(1);
			for (int i = 1;i < f.length;++ i) {
				dp[i] = dp[i - 1].multiply(x - i - 1);
				dp2[f.length - i - 1] = dp2[f.length - i].multiply(x - f.length + i);
			}
			for (int i = 0;i < f.length;++ i) {
				ModInteger tmp = f[i].multiply(dp[i]).multiplyEqual(dp2[i]).multiplyEqual(inv[i]).multiplyEqual(inv[f.length - 1 - i]);
				if ((f.length - i & 1) == 0) ret.addEqual(tmp);
				else ret.subtractEqual(tmp);
			}
			return ret;
		}
	}

	public static class AbstractArray<T> extends AbstractList<T> implements RandomAccess{

		private final Object[] array;

		public AbstractArray(int size) {
			array = new Object[size];
		}

		public AbstractArray(T[] array) {
			this(array.length);
			System.arraycopy(array, 0, this.array, 0, array.length);
		}

		@Override
		public T set(int index, T element) {
			T ret = get(index);
			array[index] = element;
			return ret;
		}

		@Override
		public T get(int index) {
			@SuppressWarnings("unchecked")
			T ret = (T)array[index];
			return ret;
		}

		public Object[] get() {
			return array;
		}

		public T[] get(T[] array) {
			if (array.length < this.array.length) {
				@SuppressWarnings("unchecked")
				T[] ret  = (T[])Arrays.copyOfRange(this.array, 0, this.array.length, array.getClass());
				return ret;
			}
			System.arraycopy(this.array, 0, array, 0, this.array.length);
			return array;
		}

		@Override
		public int size() {
			return array.length;
		}

		public int length() {
			return size();
		}

		@Override
		public int hashCode() {
			return Arrays.hashCode(array);
		}

		private class Iter implements Iterator<T> {
			private int index;

			private Iter() {
				index = 0;
			}

			@Override
			public boolean hasNext() {
				return index < array.length;
			}

			@Override
			public T next() {
				return get(index++);
			}

			@Override
			public void remove() {
				throw new UnsupportedOperationException();
			}
		}

		@Override
		public Iterator<T> iterator() {
			return new Iter();
		}
	}

	public static class Array<T> extends AbstractArray<T> implements Serializable{

		private static final long serialVersionUID = 2749604433067098063L;

		public Array(int size) {
			super(size);
		}

		public Array(T[] array) {
			super(array);
		}

		public T front() {
			return get(0);
		}

		public T back() {
			return get(size() - 1);
		}
	}

	public static abstract class Enumeration<V> implements Iterator<Enumeration.Entry<V>>, Iterable<Enumeration.Entry<V>> {

		public static class Entry<V> {
			public final int index;
			public final V value;

			public Entry(int index, V value) {
				this.index = index;
				this.value = value;
			}

			public int getIndex() {
				return index;
			}

			public V getValue() {
				return value;
			}

			@Override
			public String toString() {
				return index + ":" + value.toString();
			}
		}

		@Override
		public Iterator<Entry<V>> iterator() {
			return this;
		}

		public static <V> Enumeration<V> create(V[] array) {
			return new EnumerationArray<>(array);
		}

		public static <V> Enumeration<V> create(Collection<? extends V> collection) {
			return new EnumerationCollection<>(collection);
		}

		private static class EnumerationArray<V> extends Enumeration<V> {
			private int index;
			private final V[] array;

			public EnumerationArray(V[] array) {
				index = 0;
				this.array = array;
			}
			@Override
			public boolean hasNext() {
				return index != array.length;
			}

			@Override
			public Entry<V> next() {
				Entry<V> ret = new Entry<>(index, array[index]);
				++ index;
				return ret;
			}
		}

		private static class EnumerationCollection<V> extends Enumeration<V> {
			private int index;
			private final Iterator<? extends V> iterator;

			public EnumerationCollection(Collection<? extends V> collection) {
				index = 0;
				iterator = collection.iterator();
			}
			@Override
			public boolean hasNext() {
				return iterator.hasNext();
			}

			@Override
			public Entry<V> next() {
				Entry<V> ret = new Entry<>(index, iterator.next());
				++ index;
				return ret;
			}
		}
	}
}