Problem:
Score : 200 points

Problem Statement

In a factory, there are N robots placed on a number line.
Robot i is placed at coordinate X_i and can extend its arms of length L_i in both directions, positive and negative.

We want to remove zero or more robots so that the movable ranges of arms of no two remaining robots intersect.
Here, for each i (1 \leq i \leq N), the movable range of arms of Robot i is the part of the number line between the coordinates X_i - L_i and X_i + L_i, excluding the endpoints.

Find the maximum number of robots that we can keep.

Constraints

1 \leq N \leq 100,000

0 \leq X_i \leq 10^9 (1 \leq i \leq N)

1 \leq L_i \leq 10^9 (1 \leq i \leq N)

If i \neq j, X_i \neq X_j.

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
X_1 L_1
X_2 L_2
\vdots
X_N L_N

Output

Print the maximum number of robots that we can keep.

Sample Input 1

4
2 4
4 3
9 3
100 5

Sample Output 1

3

By removing Robot 2, we can keep the other three robots.

Sample Input 2

2
8 20
1 10

Sample Output 2

1

Sample Input 3

5
10 1
2 1
4 1
6 1
8 1

Sample Output 3

5

Solution in Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)





import java.awt.Point;
import java.io.File;
import java.io.FileInputStream;
import java.io.FileNotFoundException;
import java.io.FileOutputStream;
import java.io.IOException;
import java.io.InputStream;
import java.io.PrintStream;
import java.io.PrintWriter;
import java.io.Serializable;
import java.util.AbstractList;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.Collection;
import java.util.Comparator;
import java.util.HashMap;
import java.util.Iterator;
import java.util.List;
import java.util.Locale;
import java.util.Map;
import java.util.Map.Entry;
import java.util.NoSuchElementException;
import java.util.RandomAccess;
import java.util.Set;
import java.util.function.BinaryOperator;
import java.util.function.UnaryOperator;

public class Main implements Runnable{

	private void solve(FastIO io, String[] args) {
		/*
		 * author: 31536000
		 * キーエンスプログラミングコンテスト2020 B問題
		 * 考察メモ
		 * 区間で考えていく
		 * 各ロボットは[X-L, X+L)の区間を持ち、仮に[L, R)とする
		 * この時、この問題は重ならない区間の最大化になる
		 * dp[i]:座標P_i以下におけるロボットの答え
		 * とすると、各ロボットについて、dp[P_i](i>=r)にdp[P_l]+1以上でmaxを行う操作へ
		 * 座圧で
		 *
		 */
		int N = io.nextInt();
		IntRange[] range = new IntRange[N];
		for (int i = 0;i < N;++ i) {
			int X = io.nextInt(), L = io.nextInt();
			range[i] = IntRange.closed(X - L, X + L);
		}
		Integer[] comp = new Integer[2 * N];
		for (int i = 0;i < N;++ i) {
			comp[i << 1] = range[i].getClosedLower();
			comp[i << 1 | 1] = range[i].getClosedUpper();
		}
		Compress<Integer> compress = new Compress<>(comp); // 座圧
		Arrays.sort(range, (l, r) -> Integer.compare(l.getClosedUpper(), r.getClosedUpper())); // 終点ソート
		io.debugln(range);
		int rangeIndex = 0;
		int[] dp = new int[compress.size()]; // 答え
		for (int i = 1;i < dp.length;++ i) {
			dp[i] = dp[i - 1]; // まず、ロボットを使わない時
			while (rangeIndex < range.length && compress.zip(range[rangeIndex].getClosedUpper()) == i) { // 現在座標
				int get = compress.zip(range[rangeIndex++].getClosedLower());
				dp[i] = Math.max(dp[i], dp[get] + 1); // このロボットを採用するとき
			}
		}
		io.debugln(dp);
		io.println(dp[dp.length - 1]);
	}

	/** 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;
	private final String[] args;

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

	public Main(String[] args) {
		this(new FastIO(), args);
	}

	public Main(FastIO io, String... args) {
		this.io = io;
		this.args = args;
		if (DEBUG) io.setAutoFlush(true);
	}

	@Override
	public void run() {
		try {
			solve(io, args);
		} catch (Throwable e) {
			throw e;
		} finally {
			io.flush();
		}
	}

	// 以下、ライブラリ

	public class Compress<T extends Comparable<T>> {

		private Map<T, Integer> zip;
		private Object[] unzip;

		public Compress(T[] dat) {
			Arrays.sort(dat);
			zip = new HashMap<T, Integer>();
			List<T> unzip = new ArrayList<T>();
			for (int i = 0, j = 0; i < dat.length; ++i) {
				if (zip.put(dat[i], j++ ) != null) {
					zip.put(dat[i], --j - 1);
				} else unzip.add(dat[i]);
			}
			this.unzip = unzip.toArray();
		}

		public Compress(List<T> dat) {
			dat.sort(null);
			Object[] dats = dat.toArray();
			zip = new HashMap<T, Integer>();
			List<T> unzip = new ArrayList<T>();
			for (int i = 0, j = 0; i < dats.length; ++i) {
				@SuppressWarnings("unchecked")
				T push = (T)dats[i];
				if (zip.put(push, j++ ) != null) {
					zip.put(push, --j - 1);
				} else unzip.add(push);
			}
			this.unzip = unzip.toArray();
		}

		public int zip(T dat) {
			return zip.get(dat);
		}

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

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

	/**
	 * 高速な入出力を提供します。
	 * @author 31536000
	 *
	 */
	public static class FastIO {
		private InputStream in;
		private final byte[] buffer = new byte[1024];
		private int read = 0;
		private int length = 0;
		private PrintWriter out;
		private 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 final void setInputStream(InputStream in) {
			this.in = in;
		}

		public final void setInputStream(File in) {
			try {
				this.in = new FileInputStream(in);
			} catch (FileNotFoundException e) {
				e.printStackTrace();
			}
		}

		public final void setOutputStream(PrintStream out) {
			this.out = new PrintWriter(out, false);
		}

		public final void setOutputStream(File out) {
			try {
				this.out = new PrintWriter(new FileOutputStream(out), false);
			} catch (FileNotFoundException e) {
				e.printStackTrace();
			}
		}

		public final void setErrorStream(PrintStream err) {
			this.err = new PrintWriter(err, false);
		}

		public final void setErrorStream(File err) {
			try {
				this.err = new PrintWriter(new FileOutputStream(err), false);
			} catch (FileNotFoundException e) {
				e.printStackTrace();
			}
		}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

		public final Object print(Object obj) {
			if (obj != null && 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 final String print(String s) {
			out.print(s);
			if (autoFlush) flush();
			return s;
		}

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

		private final Object print(Object array, int check, String... parse) {
			if (check >= parse.length) {
				if (array != null && 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 final 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 final Object[] printf(String format, Object... args) {
			out.printf(format, args);
			if (autoFlush) flush();
			return args;
		}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

		public final boolean debug(boolean b) {
			err.print(b);
			if (autoFlush) flush();
			return b;
		}

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

		public final char debug(char c) {
			err.print(c);
			if (autoFlush) flush();
			return c;
		}

		public final char[] debug(char[] s) {
			err.print(s);
			return s;
		}

		public final double debug(double d) {
			err.print(d);
			if (autoFlush) flush();
			return d;
		}

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

		public final float debug(float f) {
			err.print(f);
			if (autoFlush) flush();
			return f;
		}

		public final int debug(int i) {
			err.print(i);
			if (autoFlush) flush();
			return i;
		}

		public final long debug(long l) {
			err.print(l);
			if (autoFlush) flush();
			return l;
		}

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

		public final String debug(String s) {
			err.print(s);
			if (autoFlush) flush();
			return s;
		}

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

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

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

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

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

		public final void debugln() {
			err.println();
			if (autoFlush) flush();
		}

		public final boolean debugln(boolean b) {
			err.println(b);
			if (autoFlush) flush();
			return b;
		}

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

		public final char debugln(char c) {
			err.println(c);
			if (autoFlush) flush();
			return c;
		}

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

		public final double debugln(double d) {
			err.println(d);
			if (autoFlush) flush();
			return d;
		}

		public final double debugln(double d, int length) {
			debug(d, length);
			debugln();
			return d;
		}

		public final float debugln(float f) {
			err.println(f);
			if (autoFlush) flush();
			return f;
		}

		public final int debugln(int i) {
			err.println(i);
			if (autoFlush) flush();
			return i;
		}

		public final long debugln(long l) {
			err.println(l);
			if (autoFlush) flush();
			return l;
		}

		public final Object debugln(Object obj) {
			debug(obj);
			debugln();
			return obj;
		}

		public final String debugln(String s) {
			err.println(s);
			if (autoFlush) flush();
			return s;
		}

		public final Object debugln(Object array, String... parse) {
			debug(array, parse);
			debugln();
			return array;
		}

		public final 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 Identity<T> extends BinaryOperator<T>{
		/**
		 * 単位元を返します。
		 * @return 単位元
		 */
		public T identity();
	}

	/**
	 * 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 = identity();
			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>, Identity<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 = identity();
			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();
		public default T add(T left, T right) {
			return getAddition().apply(left, right);
		}

		public default T multiply(T left, T right) {
			return getMultiplication().apply(left, right);
		}

		public default T additiveIdentity() {
			return getAddition().identity();
		}

		public default T multipleIdentity() {
			return getMultiplication().identity();
		}

		public default int characteristic() {
			return 0;
		}
	}

	/**
	 * 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>{

	}

	/**
	 * 演算が整域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 IntegralDomain<T, A extends Abelian<T>, M extends CommutativeMonoid<T>> extends CommutativeRing<T, A, M>{
		public boolean isDivisible(T left, T right);
		public T divide(T left, T right);
	}

	/**
	 * 演算が整閉整域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 IntegrallyClosedDomain<T, A extends Abelian<T>, M extends CommutativeMonoid<T>> extends IntegralDomain<T, A, M>{

	}

	/**
	 * 演算がGCD整域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 GCDDomain<T, A extends Abelian<T>, M extends CommutativeMonoid<T>> extends IntegrallyClosedDomain<T, A, M>{
		public T gcd(T left, T right);
		public T lcm(T left, T right);
	}

	/**
	 * 素元を提供します。
	 * @author 31536000
	 *
	 * @param <T> 演算の型
	 */
	public static class PrimeElement<T> {
		public final T element;
		public PrimeElement(T element) {
			this.element = element;
		}
	}

	public interface MultiSet<E> extends Collection<E>{
		public int add(E element, int occurrences);
		public int count(Object element);
		public Set<E> elementSet();
		public boolean remove(Object element, int occurrences);
		public int setCount(E element, int count);
		public boolean setCount(E element, int oldCount, int newCount);
	}

	/**
	 * 演算が一意分解整域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 UniqueFactorizationDomain<T, A extends Abelian<T>, M extends CommutativeMonoid<T>> extends GCDDomain<T, A, M>{
		public MultiSet<PrimeElement<T>> PrimeFactorization(T x);
	}

	/**
	 * 演算が主イデアル整域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 PrincipalIdealDomain<T, A extends Abelian<T>, M extends CommutativeMonoid<T>> extends UniqueFactorizationDomain<T, A, M> {

	}

	/**
	 * 演算がユークリッド整域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 EuclideanDomain<T, A extends Abelian<T>, M extends CommutativeMonoid<T>> extends PrincipalIdealDomain<T, A, M>{
		public T reminder(T left, T right);
	}

	/**
	 * 演算が体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 Field<T, A extends Abelian<T>, M extends Abelian<T>> extends EuclideanDomain<T, A, M>{
		@Override
		public default boolean isDivisible(T left, T right) {
			return !right.equals(additiveIdentity());
		}

		@Override
		public default T divide(T left, T right) {
			if (isDivisible(left, right)) throw new ArithmeticException("divide by Additive Identify");
			return multiply(left, getMultiplication().inverse(right));
		}

		@Override
		public default T reminder(T left, T right) {
			if (isDivisible(left, right)) throw new ArithmeticException("divide by Additive Identify");
			return additiveIdentity();
		}

		@Override
		public default T gcd(T left, T right) {
			return multipleIdentity();
		}

		@Override
		public default T lcm(T left, T right) {
			return multipleIdentity();
		}

		@Override
		public default MultiSet<PrimeElement<T>> PrimeFactorization(T x) {
			HashMultiSet<PrimeElement<T>> ret = HashMultiSet.create(1);
			ret.add(new PrimeElement<T>(x));
			return ret;
		}
	}

	public static class HashMultiSet<E> implements MultiSet<E>, Serializable{

		private static final long serialVersionUID = -8378919645386251159L;
		private final transient HashMap<E, Integer> map;
		private transient int size;

		private HashMultiSet() {
			map = new HashMap<>();
			size = 0;
		}

		private HashMultiSet(int distinctElements) {
			map = new HashMap<>(distinctElements);
			size = 0;
		}

		public static <E> HashMultiSet<E> create() {
			return new HashMultiSet<>();
		}

		public static <E> HashMultiSet<E> create(int distinctElements) {
			return new HashMultiSet<>(distinctElements);
		}

		public static <E> HashMultiSet<E> create(Iterable<? extends E> elements) {
			HashMultiSet<E> ret = new HashMultiSet<>();
			for (E i : elements) ret.map.compute(i, (v, e) -> e == null ? 1 : ++e);
			return ret;
		}

		@Override
		public int size() {
			return size;
		}

		@Override
		public boolean isEmpty() {
			return size == 0;
		}

		@Override
		public boolean contains(Object o) {
			return map.containsKey(o);
		}

		private class Iter implements Iterator<E> {

			private final Iterator<Entry<E, Integer>> iter = map.entrySet().iterator();
			private E value;
			private int count = 0;

			@Override
			public boolean hasNext() {
				if (count > 0) return true;
				if (iter.hasNext()) {
					Entry<E, Integer> entry = iter.next();
					value = entry.getKey();
					count = entry.getValue();
					return true;
				}
				return false;
			}

			@Override
			public E next() {
				-- count;
				return value;
			}

		}

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

		@Override
		public Object[] toArray() {
			Object[] ret = new Object[size];
			int read = 0;
			for (Entry<E, Integer> i : map.entrySet()) Arrays.fill(ret, read, read += i.getValue(), i.getKey());
			return ret;
		}

		@Override
		public <T> T[] toArray(T[] a) {
			Object[] src = toArray();
			if (a.length < src.length) {
				@SuppressWarnings("unchecked")
				T[] ret  = (T[])Arrays.copyOfRange(src, 0, src.length, a.getClass());
				return ret;
			}
			System.arraycopy(src, 0, a, 0, src.length);
			return a;
		}

		@Override
		public boolean add(E e) {
			add(e, 1);
			return true;
		}

		@Override
		public boolean remove(Object o) {
			return remove(o, 1);
		}

		@Override
		public boolean containsAll(Collection<?> c) {
			boolean ret = true;
			for (Object i : c) ret |= contains(i);
			return ret;
		}

		@Override
		public boolean addAll(Collection<? extends E> c) {
			boolean ret = false;
			for (E i : c) ret |= add(i);
			return ret;
		}

		@Override
		public boolean removeAll(Collection<?> c) {
			boolean ret = false;
			for (Object i : c) ret |= remove(i);
			return ret;
		}

		@Override
		public boolean retainAll(Collection<?> c) {
			return removeAll(c);
		}

		@Override
		public void clear() {
			map.clear();
			size = 0;
		}

		@Override
		public int add(E element, int occurrences) {
			size += occurrences;
			return map.compute(element, (k, v) -> v == null ? occurrences : v + occurrences) - occurrences;
		}

		@Override
		public int count(Object element) {
			return map.getOrDefault(element, 0);
		}

		@Override
		public Set<E> elementSet() {
			return map.keySet();
		}

		public Set<Entry<E, Integer>> entrySet() {
			return map.entrySet();
		}

		@Override
		public boolean remove(Object element, int occurrences) {
			try {
				@SuppressWarnings("unchecked")
				E put = (E) element;
				return map.compute(put, (k, v) -> {
					if (v == null) return null;
					if (v < occurrences) {
						size -= v;
						return null;
					}
					size -= occurrences;
					return v - occurrences;
				}) != null;
			} catch (ClassCastException E) {
				return false;
			}
		}

		@Override
		public int setCount(E element, int count) {
			Integer ret = map.put(element, count);
			int ret2 = ret == null ? 0 : ret;
			size += count - ret2;
			return ret2;
		}

		@Override
		public boolean setCount(E element, int oldCount, int newCount) {
			boolean ret = map.replace(element, oldCount, newCount);
			if (ret) size += newCount - oldCount;
			return ret;
		}

	}

	public static class ModInteger extends Number implements Field<ModInteger, Abelian<ModInteger>, Abelian<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 identity() {
				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 identity() {
				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));
			}

		}

		@Override
		public int characteristic() {
			return mod;
		}

		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;
		}

		protected ModInteger getNewInstance(ModInteger mod) {
			return new ModInteger(mod);
		}

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

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

		public ModInteger add(ModInteger n) {
			return getNewInstance(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 getNewInstance(this).subtractEqual(n);
		}

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

		public ModInteger subtract(ModInteger n) {
			return getNewInstance(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 getNewInstance(this).multiplyEqual(n);
		}

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

		public ModInteger multiply(ModInteger n) {
			return getNewInstance(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 getNewInstance(this).divideEqual(n);
		}

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

		public ModInteger divide(ModInteger n) {
			return getNewInstance(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 getNewInstance(this).powEqual(n);
		}

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

		public ModInteger pow(ModInteger n) {
			return getNewInstance(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;
		private int[] fact, inv, invfact;

		/**
		 * modを法として、演算を行います。
		 * @param mod 法とする素数
		 */
		public ModUtility(Prime mod) {
			this(mod, 2);
		}

		/**
		 * modを法として、演算を行います。
		 * @param mod 法とする素数
		 * @param calc 予め前計算しておく大きさ
		 */
		public ModUtility(Prime mod, int calc) {
			this.mod = mod.prime;
			precalc(calc);
		}

		/**
		 * calcの大きさだけ、前計算を行います。
		 * @param calc 前計算をする大きさ
		 */
		public void precalc(int calc) {
			++ 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);
			}
		}

		/**
		 * modを法とする剰余環上で振舞う整数を返します。
		 * @return modを法とする整数、初期値は0
		 */
		public ModInteger create() {
			return new ModInt();
		}

		/**
		 * modを法とする剰余環上で振舞う整数を返します。
		 * @param n 初期値
		 * @return modを法とする整数
		 */
		public ModInteger create(int n) {
			return new ModInt(n);
		}

		private class ModInt extends ModInteger {

			private static final long serialVersionUID = -2435281861935422575L;

			public ModInt() {
				super(mod);
			}

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

			public ModInt(ModInteger mod) {
				super(mod);
			}

			@Override
			protected ModInteger getNewInstance(ModInteger mod) {
				return new ModInt(mod);
			}

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

		/**
		 * modを法として、nの逆元を返します。<br>
		 * 計算量はO(log n)です。
		 * @param n 逆元を求めたい値
		 * @return 逆元
		 */
		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();
			}
		}

		/**
		 * n!を、modを法として求めた値を返します。<br>
		 * 計算量はO(n)です。
		 * @param n 階乗を求めたい値
		 * @return nの階乗をmodで割った余り
		 */
		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();
			}
		}

		/**
		 * nPkをmodで割った余りを求めます。<br>
		 * 計算量はO(n-k)です。
		 * @param n 左辺
		 * @param k 右辺
		 * @return nPkをmodで割った余り
		 */
		public int permutation(int n, int k) {
			if (n < 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;
		}

		/**
		 * nCkをmodで割った余りを求めます。<br>
		 * 計算量はO(n-k)です。
		 * @param n 左辺
		 * @param k 右辺
		 * @return nCkをmodで割った余り
		 */
		public int combination(int n, int k) {
			if (n < 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;
		}

		/**
		 * 他項係数をmodで割った余りを求めます。<br>]
		 * 計算量はO(n)です。
		 * @param n 左辺
		 * @param k 右辺、合計がn以下である必要がある
		 * @return 他項係数
		 */
		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;
		}

		/**
		 * n個からk個を選ぶ重複組み合わせnHkをmodで割った余りを求めます。<br>
		 * 計算量はO(min(n, k))です。
		 * @param n 左辺
		 * @param k 右辺
		 * @return nHkをmodで割った余り
		 */
		public int multichoose(int n, int k) {
			return combination(mod(n + k - 1), k);
		}

		/**
		 * カタラン数C(n)をmodで割った余りを求めます。<br>
		 * 計算量はO(n)です。
		 * @param n 求めたいカタラン数の番号
		 * @return カタラン数
		 */
		public int catalan(int n) {
			return divide(combination(mod(2 * n), n), mod(n + 1));
		}

		/**
		 * nのm乗をmodで割った余りを求めます。<br>
		 * 計算量はO(log m)です。
		 * @param n 床
		 * @param m 冪指数
		 * @return n^mをmodで割った余り
		 */
		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;
		}

		/**
		 * nのm乗をmodで割った余りを求めます。<br>
		 * 計算量はO(log m)です。
		 * @param n 床
		 * @param m 冪指数
		 * @return n^mをmodで割った余り
		 */
		public int pow(long n, long m) {
			return pow((int)(n % mod), (int)(n % (mod - 1)));
		}

		/**
		 * 現在のmod値のトーシェント数を返します。<br>
		 * なお、これはmod-1に等しいです。
		 * @return トーシェント数
		 */
		public int totient() {
			return mod - 1;
		}

		/**
		 * nのトーシェント数を返します。<br>
		 * 計算量はO(sqrt n)です。
		 * @param n トーシェント数を求めたい値
		 * @return nのトーシェント数
		 */
		public static int totient(int n) {
			int totient = n;
			for (int i = 2;i * i <= n;++ i) {
				if (n % i == 0) {
					totient = totient / i * (i - 1);
					while ((n %= i) % i == 0);
				}
			}
			if (n != 1) totient = totient / n * (n - 1);
			return totient;
		}

		/**
		 * nをmodで割った余りを返します。
		 * @param n 演算する値
		 * @return nをmodで割った余り
		 */
		public int mod(int n) {
			return (n %= mod) < 0 ? n + mod : n;
		}

		/**
		 * nをmodで割った余りを返します。
		 * @param n 演算する値
		 * @return nをmodで割った余り
		 */
		public int mod(long n) {
			return (int)((n %= mod) < 0 ? n + mod : n);
		}

		/**
		 * n+mをmodで割った余りを返します。
		 * @param n 足される値
		 * @param m 足す値
		 * @return n+mをmodで割った余り
		 */
		public int add(int n, int m) {
			return mod(n + m);
		}

		/**
		 * n-mをmodで割った余りを返します。
		 * @param n 引かれる値
		 * @param m 引く値
		 * @return n-mをmodで割った余り
		 */
		public int subtract(int n, int m) {
			return mod(n - m);
		}

		/**
		 * n*mをmodで割った余りを返します。
		 * @param n 掛けられる値
		 * @param m 掛ける値
		 * @return n*mをmodで割った余り
		 */
		public int multiply(int n, int m) {
			int ans = (int)((long)n * m % mod);
			return ans < 0 ? ans + mod : ans;
		}

		/**
		 * n/mをmodで割った余りを返します。
		 * @param n 割られる値
		 * @param m 割る値
		 * @return n/mをmodで割った余り
		 */
		public int divide(int n, int m) {
			return multiply(n, inverse(m));
		}

		/**
		 * fを通ることが分かっているfの要素数-1次の関数について、xの位置における値をmodで割った余りを返します。<br>
		 * 計算量はO(f)です。
		 * @param f 関数の形
		 * @param x 求める位置
		 * @return 求めたい値をmodで割った余り
		 */
		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;
		}
	}

	/**
	 * 素数を渡すためのクラスです。<br>
	 * 中身が確実に素数であることを保証するときに使ってください。
	 *
	 * @author 31536000
	 *
	 */
	public static class Prime extends Number{

		private static final long serialVersionUID = 8216169308184181643L;
		public final int prime;

		/**
		 * 素数を設定します。
		 *
		 * @param prime 素数
		 * @throws IllegalArgumentException 素数以外を渡した時
		 */
		public Prime(int prime) {
			if (!isPrime(prime)) throw new IllegalArgumentException(prime + " is not prime");
			this.prime = prime;
		}

		private static final int bases[] = {15591, 2018, 166, 7429, 8064, 16045, 10503, 4399, 1949, 1295, 2776, 3620, 560, 3128, 5212, 2657, 2300, 2021, 4652, 1471, 9336, 4018, 2398, 20462, 10277, 8028, 2213, 6219, 620, 3763, 4852, 5012, 3185, 1333, 6227, 5298, 1074, 2391, 5113, 7061, 803, 1269, 3875, 422, 751, 580, 4729, 10239, 746, 2951, 556, 2206, 3778, 481, 1522, 3476, 481, 2487, 3266, 5633, 488, 3373, 6441, 3344, 17, 15105, 1490, 4154, 2036, 1882, 1813, 467, 3307, 14042, 6371, 658, 1005, 903, 737, 1887, 7447, 1888, 2848, 1784, 7559, 3400, 951, 13969, 4304, 177, 41, 19875, 3110, 13221, 8726, 571, 7043, 6943, 1199, 352, 6435, 165, 1169, 3315, 978, 233, 3003, 2562, 2994, 10587, 10030, 2377, 1902, 5354, 4447, 1555, 263, 27027, 2283, 305, 669, 1912, 601, 6186, 429, 1930, 14873, 1784, 1661, 524, 3577, 236, 2360, 6146, 2850, 55637, 1753, 4178, 8466, 222, 2579, 2743, 2031, 2226, 2276, 374, 2132, 813, 23788, 1610, 4422, 5159, 1725, 3597, 3366, 14336, 579, 165, 1375, 10018, 12616, 9816, 1371, 536, 1867, 10864, 857, 2206, 5788, 434, 8085, 17618, 727, 3639, 1595, 4944, 2129, 2029, 8195, 8344, 6232, 9183, 8126, 1870, 3296, 7455, 8947, 25017, 541, 19115, 368, 566, 5674, 411, 522, 1027, 8215, 2050, 6544, 10049, 614, 774, 2333, 3007, 35201, 4706, 1152, 1785, 1028, 1540, 3743, 493, 4474, 2521, 26845, 8354, 864, 18915, 5465, 2447, 42, 4511, 1660, 166, 1249, 6259, 2553, 304, 272, 7286, 73, 6554, 899, 2816, 5197, 13330, 7054, 2818, 3199, 811, 922, 350, 7514, 4452, 3449, 2663, 4708, 418, 1621, 1171, 3471, 88, 11345, 412, 1559, 194};

		private static boolean isSPRP(int n, int a) {
			int d = n - 1, s = 0;
			while ((d & 1) == 0) {
				++s;
				d >>= 1;
			}
			long cur = 1, pw = d;
			while (pw != 0) {
				if ((pw & 1) != 0) cur = (cur * a) % n;
				a = (int)(((long)a * a) % n);
				pw >>= 1;
			}
			if (cur == 1) return true;
			for (int r = 0; r < s; r++ ) {
				if (cur == n - 1) return true;
				cur = (cur * cur) % n;
			}
			return false;
		}

		/**
		 * 与えられた値が素数か否かを判定します。<br>
		 * この実装はhttp://ceur-ws.org/Vol-1326/020-Forisek.pdfに基づきます。
		 * @param x 判定したい値
		 * @return xが素数ならtrue
		 */
		public static boolean isPrime(int x) {
			if (x == 2 || x == 3 || x == 5 || x == 7) return true;
			if ((x & 1) == 0 || x % 3 == 0 || x % 5 == 0 || x % 7 == 0) return false;
			if (x < 121) return x > 1;
			long h = x;
			h = ((h >> 16) ^ h) * 0x45d9f3b;
			h = ((h >> 16) ^ h) * 0x45d9f3b;
			h = ((h >> 16) ^ h) & 0xFF;
			return isSPRP(x, bases[(int)h]);
		}

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

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

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

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

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

	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);
		}
	}

	/**
	 * 要素とそのindexを効率的に取得する関数を提供します。
	 * @author 31536000
	 *
	 */
	public static class Enumeration {
		/**
		 * 要素とそのindexを管理するクラスです。
		 * @author 31536000
		 *
		 * @param <E> 保持する要素
		 */
		public static class Enumerate<E> {
			public final E value;
			public final int index;

			/**
			 * 要素とそのindexを渡します。<br>
			 * indexは必ずしも元の配列またはコレクションのindexと一致する必要はありませんが、一致する値を返すことが推奨されます。
			 * @param value
			 * @param index
			 */
			public Enumerate(E value, int index) {
				this.value = value;
				this.index = index;
			}

			/**
			 * 要素を返します。
			 * @return 要素
			 */
			public E getValue() {
				return value;
			}

			/**
			 * indexを返します。
			 * @return index
			 */
			public int getIndex() {
				return index;
			}

			@Override
			public boolean equals(Object o) {
				if (o instanceof Enumerate) return ((Enumerate<?>)o).getValue().equals(value) && ((Enumerate<?>)o).getIndex() == index;
				return false;
			}

			@Override
			public int hashCode() {
				return value.hashCode() ^ index;
			}

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

		private static class IteratorArray<E> implements Iterator<Enumerate<E>>{
			private final E[] array;
			private final int start;
			private int index;

			public IteratorArray(E[] array, int index) {
				this.array = array;
				this.start = index;
				this.index = 0;
			}

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

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

		private static class IteratorCollection<E> implements Iterator<Enumerate<E>>{
			private final Iterator<E> iter;
			private int start;

			public IteratorCollection(Iterator<E> iter, int index) {
				this.iter = iter;
				this.start = index;
			}

			@Override
			public boolean hasNext() {
				return iter.hasNext();
			}

			@Override
			public Enumerate<E> next() {
				Enumerate<E> ret = new Enumerate<>(iter.next(), start++);
				return ret;
			}
		}

		/**
		 * 配列の各要素とそのindexを順に返すIteratorを生成します。
		 * @param <E> 配列の型
		 * @param array 配列
		 * @return Enumerate&lt;E&gt;のIterator
		 */
		public static <E> Iterator<Enumerate<E>> enumerate(E[] array) {
			return enumerate(array, 0);
		}

		/**
		 * 配列の各要素とそのindexを順に返すIteratorを生成します。
		 * @param <E> 配列の型
		 * @param array 配列
		 * @param start 添字の初期値、この値だけindexが足されたものが返る
		 * @return Enumerate&lt;E&gt;のIterator
		 */
		public static <E> Iterator<Enumerate<E>> enumerate(E[] array, int start) {
			if (array == null) throw new NullPointerException("array is null");
			return new IteratorArray<E>(array, start);
		}

		/**
		 * Iteratorの各要素とそのindexを順に返すIteratorを生成します。
		 * @param <E> Iteratorの型
		 * @param iter Iterator
		 * @return Enumerate&lt;E&gt;のIterator
		 */
		public static <E> Iterator<Enumerate<E>> enumerate(Iterator<E> iter) {
			return enumerate(iter, 0);
		}

		/**
		 * Iteratorの各要素とそのindexを順に返すIteratorを生成します。
		 * @param <E> Iteratorの型
		 * @param iter Iterator
		 * @param start 添字の初期値、この値だけindexが足されたものが返る
		 * @return Enumerate&lt;E&gt;のIterator
		 */
		public static <E> Iterator<Enumerate<E>> enumerate(Iterator<E> iter, int start) {
			if (iter == null) throw new NullPointerException("iterator is null");
			return new IteratorCollection<E>(iter, start);
		}
	}
}