Problem:
Score : 100 points

Problem Statement

You are given a grid of N rows and M columns. The square at the i-th row and j-th column will be denoted as (i,j).
Some of the squares contain an object. All the remaining squares are empty.
The state of the grid is represented by strings S_1,S_2,\cdots,S_N. The square (i,j) contains an object if S_{i,j}= # and is empty if S_{i,j}= ..

Consider placing 1 \times 2 tiles on the grid. Tiles can be placed vertically or horizontally to cover two adjacent empty squares.
Tiles must not stick out of the grid, and no two different tiles may intersect. Tiles cannot occupy the square with an object.

Calculate the maximum number of tiles that can be placed and any configulation that acheives the maximum.

Constraints

1 \leq N \leq 100

1 \leq M \leq 100

S_i is a string with length M consists of # and ..

Input

Input is given from Standard Input in the following format:

N M
S_1
S_2
\vdots
S_N

Output

On the first line, print the maximum number of tiles that can be placed.

On the next N lines, print a configulation that achieves the maximum.
Precisely, output the strings t_1,t_2,\cdots,t_N constructed by the following way.

t_i is initialized to S_i.

For each (i,j), if there is a tile that occupies (i,j) and (i+1,j), change t_{i,j}:=v, t_{i+1,j}:=^.

For each (i,j), if there is a tile that occupies (i,j) and (i,j+1), change t_{i,j}:=>, t_{i,j+1}:=<.

See samples for further information.

You may print any configulation that maximizes the number of tiles.

Sample Input 1

3 3
#..
..#
...

Sample Output 1

3
#><
vv#
^^.

The following output is also treated as a correct answer.

3
#><
v.#
^><

Solution in Java (OpenJDK 11.0.6) (judge verdict: Accepted)

import java.util.*;
import java.io.*;
import java.math.*;
import java.util.function.*;
public class Main implements Runnable {
	static boolean DEBUG;
	public static void main(String[] args) {
		DEBUG = args.length > 0 && args[0].equals("-DEBUG");
		Thread.setDefaultUncaughtExceptionHandler((t, e) -> { e.printStackTrace(); System.exit(1); });
		new Thread(null, new Main(), "", 1 << 31).start();
	}

	public void run() {
		Solver solver = new Solver();
		solver.solve();
		solver.exit();
	}

	static class FastScanner {
		private final InputStream in = System.in;
		private final byte[] buffer = new byte[1024];
		private int pointer = 0;
		private int buflen = 0;
		private boolean hasNextByte() {
			if(pointer < buflen) return true;
			else {
				pointer = 0;
				try {
					buflen = in.read(buffer);
				}catch (IOException e) {
					e.printStackTrace();
				}
				return buflen > 0;
			}
		}
		private int readByte() { if(hasNextByte()) return buffer[pointer ++]; else return -1; }
		private boolean isPrintableChar(int c) { return isPrintableChar(c, false); }
		private boolean isPrintableChar(int c, boolean includingSpace) { return (includingSpace ? 32 : 33) <= c && c <= 126; }
		private void skipUnprintable() { skipUnprintable(false); }
		private void skipUnprintable(boolean includingSpace) { while(hasNextByte() && !isPrintableChar(buffer[pointer], includingSpace)) pointer++; }
		private boolean hasNext() { return hasNext(false); }
		private boolean hasNext(boolean includingSpace) { skipUnprintable(includingSpace); return hasNextByte(); }
		private StringBuilder sb = new StringBuilder();
		public String next() { return next(false); }
		public String next(boolean includingSpace) {
			if(!hasNext(includingSpace)) throw new NoSuchElementException();
			sb.setLength(0);
			int b = readByte();
			while(isPrintableChar(b, includingSpace)) {
				sb.appendCodePoint(b);
				b = readByte();
			}
			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(b < '0' || '9' < b) throw new NumberFormatException();
			while(true) {
				if('0' <= b && b <= '9') n = n * 10 + b - '0';
				else if(b == -1 || !isPrintableChar(b)) return minus ? -n : n;
				else throw new NumberFormatException();
				b = readByte();
			}
		}
	}

	static class Solver {
		FastScanner sc = new FastScanner();
		public Solver() { }

		String ns() { return ns(false); }
		String ns(boolean includingSpace) { return sc.next(includingSpace); }
		String[] ns(int n) { return ns(n, false); }
		String[] ns(int n, boolean includingSpace) {
			String a[] = new String[n];
			for(int i = 0; i < n; i ++) a[i] = ns(includingSpace);
			return a;
		}
		String[][] ns(int n, int m) { return ns(n, m, false); }
		String[][] ns(int n, int m, boolean includingSpace) {
			String a[][] = new String[n][m];
			for(int i = 0; i < n; i ++) a[i] = ns(m, includingSpace);
			return a;
		}
		char nc() { return ns().charAt(0); }
		char[] nc(int n) {
			String str = ns();
			if(n < 0) n = str.length();
			char a[] = new char[n];
			for(int i = 0; i < n; i ++) a[i] = str.charAt(i);
			return a;
		}
		char[][] nc(int n, int m) {
			char a[][] = new char[n][m];
			for(int i = 0; i < n; i ++) a[i] = nc(m);
			return a;
		}
		boolean[] nb(int n, char t) {
			char c[] = nc(-1);
			if(n < 0) n = c.length;
			boolean a[] = new boolean[n];
			for(int i = 0; i < n; i ++) a[i] = c[i] == t;
			return a;
		}
		boolean[][] nb(int n, int m, char t) {
			boolean a[][] = new boolean[n][m];
			for(int i = 0; i < n; i ++) a[i] = nb(m, t);
			return a;
		}
		int ni() { return Math.toIntExact(sc.nextLong()); }
		int[] ni(int n) {
			int a[] = new int[n];
			for(int i = 0; i < n; i ++) a[i] = ni();
			return a;
		}
		int[][] ni(int n, int m) {
			int a[][] = new int[n][m];
			for(int i = 0; i < n; i ++) a[i] = ni(m);
			return a;
		}
		long nl() { return sc.nextLong(); }
		long[] nl(int n) {
			long a[] = new long[n];
			for(int i = 0; i < n; i ++) a[i] = nl();
			return a;
		}
		long[][] nl(int n, int m) {
			long a[][] = new long[n][m];
			for(int i = 0; i < n; i ++) a[i] = nl(m);
			return a;
		}
		double nd() { return Double.parseDouble(sc.next()); }
		double[] nd(int n) {
			double a[] = new double[n];
			for(int i = 0; i < n; i ++) a[i] = nd();
			return a;
		}
		double[][] nd(int n, int m) {
			double a[][] = new double[n][m];
			for(int i = 0; i < n; i ++) a[i] = nd(m);
			return a;
		}		PairII npii() { return new PairII(ni(), ni()); }
		PairII[] npii(int n) {
			PairII a[] = new PairII[n];
			for(int i = 0; i < n; i ++) a[i] = npii();
			return a;
		}
		PairII[][] npii(int n, int m) {
			PairII a[][] = new PairII[n][m];
			for(int i = 0; i < n; i ++) a[i] = npii(m);
			return a;
		}
		PairIL npil() { return new PairIL(ni(), nl()); }
		PairIL[] npil(int n) {
			PairIL a[] = new PairIL[n];
			for(int i = 0; i < n; i ++) a[i] = npil();
			return a;
		}
		PairIL[][] npil(int n, int m) {
			PairIL a[][] = new PairIL[n][m];
			for(int i = 0; i < n; i ++) a[i] = npil(m);
			return a;
		}
		PairID npid() { return new PairID(ni(), nd()); }
		PairID[] npid(int n) {
			PairID a[] = new PairID[n];
			for(int i = 0; i < n; i ++) a[i] = npid();
			return a;
		}
		PairID[][] npid(int n, int m) {
			PairID a[][] = new PairID[n][m];
			for(int i = 0; i < n; i ++) a[i] = npid(m);
			return a;
		}
		PairLI npli() { return new PairLI(nl(), ni()); }
		PairLI[] npli(int n) {
			PairLI a[] = new PairLI[n];
			for(int i = 0; i < n; i ++) a[i] = npli();
			return a;
		}
		PairLI[][] npli(int n, int m) {
			PairLI a[][] = new PairLI[n][m];
			for(int i = 0; i < n; i ++) a[i] = npli(m);
			return a;
		}
		PairLL npll() { return new PairLL(nl(), nl()); }
		PairLL[] npll(int n) {
			PairLL a[] = new PairLL[n];
			for(int i = 0; i < n; i ++) a[i] = npll();
			return a;
		}
		PairLL[][] npll(int n, int m) {
			PairLL a[][] = new PairLL[n][m];
			for(int i = 0; i < n; i ++) a[i] = npll(m);
			return a;
		}
		PairLD npld() { return new PairLD(nl(), nd()); }
		PairLD[] npld(int n) {
			PairLD a[] = new PairLD[n];
			for(int i = 0; i < n; i ++) a[i] = npld();
			return a;
		}
		PairLD[][] npld(int n, int m) {
			PairLD a[][] = new PairLD[n][m];
			for(int i = 0; i < n; i ++) a[i] = npld(m);
			return a;
		}
		PairDI npdi() { return new PairDI(nd(), ni()); }
		PairDI[] npdi(int n) {
			PairDI a[] = new PairDI[n];
			for(int i = 0; i < n; i ++) a[i] = npdi();
			return a;
		}
		PairDI[][] npdi(int n, int m) {
			PairDI a[][] = new PairDI[n][m];
			for(int i = 0; i < n; i ++) a[i] = npdi(m);
			return a;
		}
		PairDL npdl() { return new PairDL(nd(), nl()); }
		PairDL[] npdl(int n) {
			PairDL a[] = new PairDL[n];
			for(int i = 0; i < n; i ++) a[i] = npdl();
			return a;
		}
		PairDL[][] npdl(int n, int m) {
			PairDL a[][] = new PairDL[n][m];
			for(int i = 0; i < n; i ++) a[i] = npdl(m);
			return a;
		}
		PairDD npdd() { return new PairDD(nd(), nd()); }
		PairDD[] npdd(int n) {
			PairDD a[] = new PairDD[n];
			for(int i = 0; i < n; i ++) a[i] = npdd();
			return a;
		}
		PairDD[][] npdd(int n, int m) {
			PairDD a[][] = new PairDD[n][m];
			for(int i = 0; i < n; i ++) a[i] = npdd(m);
			return a;
		}

		String booleanToString(boolean b) { return b ? "#" : "."; }

		PrintWriter out = new PrintWriter(System.out);
		PrintWriter err = new PrintWriter(System.err);
		StringBuilder sb4prtln = new StringBuilder();
		void prt() { out.print(""); }
		<T> void prt(T a) { out.print(a); }
		void prtln() { out.println(""); }
		<T> void prtln(T a) { out.println(a); }
		void prtln(int... a) {
			sb4prtln.setLength(0);
			for(int element : a) sb4prtln.append(element+" ");
			prtln(sb4prtln.toString().trim());
		}
		void prtln(long... a) {
			sb4prtln.setLength(0);
			for(long element : a) sb4prtln.append(element+" ");
			prtln(sb4prtln.toString().trim());
		}
		void prtln(double... a) {
			sb4prtln.setLength(0);
			for(double element : a) sb4prtln.append(element+" ");
			prtln(sb4prtln.toString().trim());
		}
		void prtln(String... a) {
			sb4prtln.setLength(0);
			for(String element : a) sb4prtln.append(element+" ");
			prtln(sb4prtln.toString().trim());
		}
		void prtln(char... a) {
			sb4prtln.setLength(0);
			for(char element : a) sb4prtln.append(element);
			prtln(sb4prtln.toString());
		}
		void prtln(boolean... a) {
			sb4prtln.setLength(0);
			for(boolean element : a) sb4prtln.append(booleanToString(element));
			prtln(sb4prtln.toString());
		}
		void prtln(int[][] a) { for(int[] element : a) prtln(element); }
		void prtln(long[][] a) { for(long[] element : a) prtln(element); }
		void prtln(double[][] a) { for(double[] element : a) prtln(element); }
		void prtln(String[][] a) { for(String[] element : a) prtln(element); }
		void prtln(char[][] a) { for(char[] element : a) prtln(element); }
		void prtln(boolean[][] a) { for(boolean[] element : a) prtln(element); }

		String errconvert(int a) { return isINF(a) ? "_" : String.valueOf(a); }
		String errconvert(long a) { return isINF(a) ? "_" : String.valueOf(a); }
		void errprt(int a) { if(DEBUG) err.print(errconvert(a)); }
		void errprt(long a) { if(DEBUG) err.print(errconvert(a)); }
		void errprt() { if(DEBUG) err.print(""); }
		<T> void errprt(T a) { if(DEBUG) err.print(a); }
		void errprt(boolean a) { if(DEBUG) errprt(booleanToString(a)); }
		void errprtln() { if(DEBUG) err.println(""); }
		void errprtln(int a) { if(DEBUG) err.println(errconvert(a)); }
		void errprtln(long a) { if(DEBUG) err.println(errconvert(a)); }
		<T> void errprtln(T a) { if(DEBUG) err.println(a); }
		void errprtln(boolean a) { if(DEBUG) errprtln(booleanToString(a)); }
		void errprtln(int... a) {
			if(DEBUG) {
				sb4prtln.setLength(0);
				for(int element : a) sb4prtln.append(errconvert(element)+" ");
				errprtln(sb4prtln.toString().trim());
			}
		}
		void errprtln(long... a) {
			if(DEBUG) {
				sb4prtln.setLength(0);
				for(long element : a) sb4prtln.append(errconvert(element)+" ");
				errprtln(sb4prtln.toString().trim());
			}
		}
		void errprtln(double... a) {
			if(DEBUG) {
				sb4prtln.setLength(0);
				for(double element : a) sb4prtln.append(element+" ");
				errprtln(sb4prtln.toString().trim());
			}
		}
		void errprtln(String... a) {
			if(DEBUG) {
				sb4prtln.setLength(0);
				for(String element : a) sb4prtln.append(element+" ");
				errprtln(sb4prtln.toString().trim());
			}
		}
		void errprtln(char... a) {
			if(DEBUG) {
				sb4prtln.setLength(0);
				for(char element : a) sb4prtln.append(element);
				errprtln(sb4prtln.toString());
			}
		}
		void errprtln(boolean... a) {
			if(DEBUG) {
				sb4prtln.setLength(0);
				for(boolean element : a) sb4prtln.append(booleanToString(element));
				errprtln(sb4prtln.toString());
			}
		}
		void errprtln(Object[] a) { if(DEBUG) for(Object element : a) errprtln(element); }
		void errprtln(int[][] a) { if(DEBUG) for(int[] element : a) errprtln(element); }
		void errprtln(long[][] a) { if(DEBUG) for(long[] element : a) errprtln(element); }
		void errprtln(double[][] a) { if(DEBUG) for(double[] element : a) errprtln(element); }
		void errprtln(String[][] a) { if(DEBUG) for(String[] element : a) errprtln(element); }
		void errprtln(char[][] a) { if(DEBUG) for(char[] element : a) errprtln(element); }
		void errprtln(boolean[][] a) { if(DEBUG) for(boolean[] element : a) errprtln(element); }
		void errprtln(Object[][] a) { if(DEBUG) for(Object element : a) { errprtln(element); errprtln(); } }

		void reply(boolean b) { prtln(b ? "Yes" : "No"); }
		void REPLY(boolean b) { prtln(b ? "YES" : "NO"); }

		void flush() { out.flush(); if(DEBUG) err.flush(); }
		void assertion(boolean b) { if(!b) { flush(); throw new AssertionError(); } }

		void exit() { flush(); System.exit(0); }
		<T> void exit(T a) { prtln(a); exit(); }
		void exit(int... a) { prtln(a); exit(); }
		void exit(long... a) { prtln(a); exit(); }
		void exit(double... a) { prtln(a); exit(); }
		void exit(String... a) { prtln(a); exit(); }
		void exit(char... a) { prtln(a); exit(); }
		void exit(boolean... a) { prtln(a); exit(); }
		void exit(int[][] a) { prtln(a); exit(); }
		void exit(long[][] a) { prtln(a); exit(); }
		void exit(double[][] a) { prtln(a); exit(); }
		void exit(String[][] a) { prtln(a); exit(); }
		void exit(char[][] a) { prtln(a); exit(); }
		void exit(boolean[][] a) { prtln(a); exit(); }


		final long INF = (long)1e18 + 7;
		boolean isPlusINF(long a) { return a > INF / 10; }
		boolean isMinusINF(long a) { return isPlusINF(- a); }
		boolean isINF(long a) { return isPlusINF(a) || isMinusINF(a); }
		final int I_INF = (int)1e9 + 7;
		boolean isPlusINF(int a) { return a > I_INF / 10; }
		boolean isMinusINF(int a) { return isPlusINF(- a); }
		boolean isINF(int a) { return isPlusINF(a) || isMinusINF(a); }


		int min(int a, int b) { return Math.min(a, b); }
		long min(long a, long b) { return Math.min(a, b); }
		double min(double a, double b) { return Math.min(a, b); }
		<T extends Comparable<T>> T min(T a, T b) { return a.compareTo(b) <= 0 ? a : b; }
		int min(int... x) {
			int min = x[0];
			for(int val : x) min = min(min, val);
			return min;
		}
		long min(long... x) {
			long min = x[0];
			for(long val : x) min = min(min, val);
			return min;
		}
		double min(double... x) {
			double min = x[0];
			for(double val : x) min = min(min, val);
			return min;
		}
		int max(int a, int b) { return Math.max(a, b); }
		long max(long a, long b) { return Math.max(a, b); }
		double max(double a, double b) { return Math.max(a, b); }
		<T extends Comparable<T>> T max(T a, T b) { return a.compareTo(b) >= 0 ? a : b; }
		int max(int... x) {
			int max = x[0];
			for(int val : x) max = max(max, val);
			return max;
		}
		long max(long... x) {
			long max = x[0];
			for(long val : x) max = max(max, val);
			return max;
		}
		double max(double... x) {
			double max = x[0];
			for(double val : x) max = max(max, val);
			return max;
		}
		<T extends Comparable<T>> T max(T[] x) {
			T max = x[0];
			for(T val : x) max = max(max, val);
			return max;
		}
		int max(int[][] a) {
			int max = a[0][0];
			for(int[] ele : a) max = max(max, max(ele));
			return max;
		}
		long max(long[][] a) {
			long max = a[0][0];
			for(long[] ele : a) max = max(max, max(ele));
			return max;
		}
		double max(double[][] a) {
			double max = a[0][0];
			for(double[] ele : a) max = max(max, max(ele));
			return max;
		}
		<T extends Comparable<T>> T max(T[][] a) {
			T max = a[0][0];
			for(T[] ele : a) max = max(max, max(ele));
			return max;
		}
		long sum(int... a) {
			long sum = 0;
			for(int element : a) sum += element;
			return sum;
		}
		long sum(long... a) {
			long sum = 0;
			for(long element : a) sum += element;
			return sum;
		}
		double sum(double... a) {
			double sum = 0;
			for(double element : a) sum += element;
			return sum;
		}
		long sum(boolean... a) {
			long sum = 0;
			for(boolean element : a) sum += element ? 1 : 0;
			return sum;
		}
		long[] sums(int[] a) {
			long sum[] = new long[a.length + 1];
			sum[0] = 0;
			for(int i = 0; i < a.length; i ++) sum[i + 1] = sum[i] + a[i];
			return sum;
		}
		long[] sums(long[] a) {
			long sum[] = new long[a.length + 1];
			sum[0] = 0;
			for(int i = 0; i < a.length; i ++) sum[i + 1] = sum[i] + a[i];
			return sum;
		}
		double[] sums(double[] a) {
			double sum[] = new double[a.length + 1];
			sum[0] = 0;
			for(int i = 0; i < a.length; i ++) sum[i + 1] = sum[i] + a[i];
			return sum;
		}
		long[] sums(boolean[] a) {
			long sum[] = new long[a.length + 1];
			sum[0] = 0;
			for(int i = 0; i < a.length; i ++) sum[i + 1] = sum[i] + (a[i] ? 1 : 0);
			return sum;
		}
		long[][] sums(int[][] a) {
			long sum[][] = new long[a.length + 1][a[0].length + 1];
			fill(sum, 0);
			for(int i = 0; i < a.length; i ++) {
				for(int j = 0; j < a[i].length; j ++) {
					sum[i + 1][j + 1] = sum[i + 1][j] + sum[i][j + 1] - sum[i][j] + a[i][j];
				}
			}
			return sum;
		}
		long[][] sums(long[][] a) {
			long sum[][] = new long[a.length + 1][a[0].length + 1];
			fill(sum, 0);
			for(int i = 0; i < a.length; i ++) {
				for(int j = 0; j < a[i].length; j ++) {
					sum[i + 1][j + 1] = sum[i + 1][j] + sum[i][j + 1] - sum[i][j] + a[i][j];
				}
			}
			return sum;
		}
		double[][] sums(double[][] a) {
			double sum[][] = new double[a.length + 1][a[0].length + 1];
			fill(sum, 0);
			for(int i = 0; i < a.length; i ++) {
				for(int j = 0; j < a[i].length; j ++) {
					sum[i + 1][j + 1] = sum[i + 1][j] + sum[i][j + 1] - sum[i][j] + a[i][j];
				}
			}
			return sum;
		}
		long[][] sums(boolean[][] a) {
			long sum[][] = new long[a.length + 1][a[0].length + 1];
			fill(sum, 0);
			for(int i = 0; i < a.length; i ++) {
				for(int j = 0; j < a[i].length; j ++) {
					sum[i + 1][j + 1] = sum[i + 1][j] + sum[i][j + 1] - sum[i][j] + (a[i][j] ? 1 : 0);
				}
			}
			return sum;
		}
		int constrain(int x, int l, int r) { return min(max(x, min(l, r)), max(l, r)); }
		long constrain(long x, long l, long r) { return min(max(x, min(l, r)), max(l, r)); }
		double constrain(double x, double l, double r) { return min(max(x, min(l, r)), max(l, r)); }
		int abs(int x) { return x >= 0 ? x : - x; }
		long abs(long x) { return x >= 0 ? x : - x; }
		double abs(double x) { return x >= 0 ? x : - x; }
		int signum(int x) { return x > 0 ? 1 : x < 0 ? -1 : 0; }
		int signum(long x) { return x > 0 ? 1 : x < 0 ? -1 : 0; }
		int signum(double x) { return x > 0 ? 1 : x < 0 ? -1 : 0; }
		long round(double x) { return Math.round(x); }
		long floor(double x) { return (long)Math.floor(x); }
		int divfloor(int a, int b) { return signum(a) == signum(b) ? a / b : - divceil(abs(a), abs(b)); }
		long divfloor(long a, long b) { return signum(a) == signum(b) ? a / b : - divceil(abs(a), abs(b)); }
		long ceil(double x) { return (long)Math.ceil(x); }
		int divceil(int a, int b) { return a >= 0 && b > 0 ? (a + b - 1) / b
											: a < 0 && b < 0 ? divceil(abs(a), abs(b))
											: - divfloor(abs(a), abs(b)); }
		long divceil(long a, long b) { return a >= 0 && b > 0 ? (a + b - 1) / b
											: a < 0 && b < 0 ? divceil(abs(a), abs(b))
											: - divfloor(abs(a), abs(b)); }
		double sqrt(int x) { return Math.sqrt((double)x); }
		double sqrt(long x) { return Math.sqrt((double)x); }
		double sqrt(double x) { return Math.sqrt(x); }
		long fact(int n) {
			long ans = 1;
			for(int i = 1; i <= n; i ++) ans = Math.multiplyExact(ans, i);
			return ans;
		}
		double pow(double x, double y) { return Math.pow(x, y); }
		long pow(long x, long y) {
			long ans = 1;
			while(true) {
				if(y % 2 != 0) ans = Math.multiplyExact(ans, x);
				y /= 2;
				if(y <= 0) return ans;
				x = Math.multiplyExact(x, x);
			}
		}
		int gcd(int a, int b) {
			while(true) {
				if(b == 0) return a;
				int tmp = a;
				a = b;
				b = tmp % b;
			}
		}
		long gcd(long a, long b) {
			while(true) {
				if(b == 0) return a;
				long tmp = a;
				a = b;
				b = tmp % b;
			}
		}
		long lcm(long a, long b) { return a / gcd(a, b) * b; }
		int gcd(int... a) {
			int gcd = 0;
			for(int i = 0; i < a.length; i ++) gcd = gcd(gcd, a[i]);
			return gcd;
		}
		long gcd(long... a) {
			long gcd = 0;
			for(int i = 0; i < a.length; i ++) gcd = gcd(gcd, a[i]);
			return gcd;
		}
		double random() { return Math.random(); }
		int random(int max) { return (int)floor(random() * max); }
		long random(long max) { return floor(random() * max); }
		double random(double max) { return random() * max; }
		int random(int min, int max) { return random(max - min) + min; }
		long random(long min, long max) { return random(max - min) + min; }
		double random(double min, double max) { return random(max - min) + min; }

		boolean isUpper(char a) { return a >= 'A' && a <= 'Z'; }
		boolean isLower(char a) { return a >= 'a' && a <= 'z'; }
		int upperToInt(char a) { return a - 'A'; }
		int lowerToInt(char a) { return a - 'a'; }
		int numToInt(char a) { return a - '0'; }
		int charToInt(char a) { return a >= 'a' ? lowerToInt(a) : a >= 'A' ? upperToInt(a) : numToInt(a); }
		char intToUpper(int a) { return (char)(a + 'A'); }
		char intToLower(int a) { return (char)(a + 'a'); }
		char intToNum(int a) { return (char)(a + '0'); }
		int[] charToInt(char[] a) {
			int array[] = new int[a.length];
			for(int i = 0; i < a.length; i ++) array[i] = charToInt(a[i]);
			return array;
		}

		long[] div(long a) {
			List<Long> divList = new ArrayList<Long>();
			for(long i = 1; i * i <= a; i ++) {
				if(a % i == 0) {
					divList.add(i);
					if(i * i != a) divList.add(a / i);
				}
			}
			long div[] = new long[divList.size()];
			for(int i = 0; i < divList.size(); i ++) div[i] = divList.get(i);
			Arrays.sort(div);
			return div;
		}

		long[][] factor(long a) {
			List<Long> factorList = new ArrayList<Long>();
			List<Long> degreeList = new ArrayList<Long>();
			for(long i = 2; i * i <= a; i ++) {
				if(a % i == 0) {
					long count = 0;
					while(a % i == 0) { a /= i; count ++; }
					factorList.add(i);
					degreeList.add(count);
				}
			}
			if(a > 1) { factorList.add(a); degreeList.add(1L); }
			long factor[][] = new long[factorList.size()][2];
			for(int i = 0; i < factorList.size(); i ++) {
				factor[i][0] = factorList.get(i);
				factor[i][1] = degreeList.get(i);
			}
			Arrays.sort(factor, (sort1, sort2) -> Long.compare(sort1[0], sort2[0]));
			return factor;
		}

		boolean isPrime(long x) {
			boolean ok = x > 1;
			for(long i = 2; i * i <= x; i ++) {
				ok &= x % i != 0;
				if(!ok) return ok;
			}
			return ok;
		}
		boolean[] prime(int num) {
			boolean prime[] = new boolean[num];
			fill(prime, true);
			if(num > 0) prime[0] = false;
			if(num > 1) prime[1] = false;
			for(int i = 2; i < num; i ++) if(prime[i]) for(int j = 2; i * j < num; j ++) prime[i * j] = false;
			return prime;
		}

		long[][] countElements(long[] a, boolean sort) {
			int len = a.length;
			long array[] = new long[len];
			for(int i = 0; i < len; i ++) array[i] = a[i];
			if(sort) Arrays.sort(array);
			List<Long> elem = new ArrayList<Long>();
			List<Long> cnt = new ArrayList<Long>();
			long tmp = 1;
			for(int i = 1; i <= len; i ++) {
				if(i == len || array[i] != array[i - 1]) {
					elem.add(array[i - 1]);
					cnt.add(tmp);
					tmp = 1;
				}else tmp ++;
			}
			long counts[][] = new long[elem.size()][2];
			for(int i = 0; i < elem.size(); i ++) {
				counts[i][0] = elem.get(i);
				counts[i][1] = cnt.get(i);
			}
			return counts;
		}
		long[][] countElements(String str, boolean sort) {
			int len = str.length();
			char array[] = str.toCharArray();
			if(sort) Arrays.sort(array);
			List<Long> elem = new ArrayList<Long>();
			List<Long> cnt = new ArrayList<Long>();
			long tmp = 1;
			for(int i = 1; i <= len; i ++) {
				if(i == len || array[i] != array[i - 1]) {
					elem.add((long)array[i - 1]);
					cnt.add(tmp);
					tmp = 1;
				}else tmp ++;
			}
			long counts[][] = new long[elem.size()][2];
			for(int i = 0; i < elem.size(); i ++) {
				counts[i][0] = elem.get(i);
				counts[i][1] = cnt.get(i);
			}
			return counts;
		}

		int[] baseConvert(long x, int n, int len) {
			int digit[] = new int[len];
			int i = 0;
			long tmp = x;
			while(tmp > 0 && i < len) { digit[i ++] = (int)(tmp % n); tmp /= n; }
			return digit;
		}
		int[] baseConvert(long x, int n) {
			long tmp = x;
			int len = 0;
			while(tmp > 0) { tmp /= n; len ++; }
			return baseConvert(x, n, len);
		}
		int[] baseConvert(int x, int n, int len) {
			int digit[] = new int[len];
			int i = 0;
			int tmp = x;
			while(tmp > 0 && i < len) { digit[i ++] = (int)(tmp % n); tmp /= n; }
			return digit;
		}
		int[] baseConvert(int x, int n) {
			int tmp = x;
			int len = 0;
			while(tmp > 0) { tmp /= n; len ++; }
			return baseConvert(x, n, len);
		}
		long[] baseConvert(long x, long n, int len) {
			long digit[] = new long[len];
			int i = 0;
			long tmp = x;
			while(tmp > 0 && i < len) { digit[i ++] = tmp % n; tmp /= n; }
			return digit;
		}
		long[] baseConvert(long x, long n) {
			long tmp = x;
			int len = 0;
			while(tmp > 0) { tmp /= n; len ++; }
			return baseConvert(x, n, len);
		}

		int numDigits(long a) { return Long.toString(a).length(); }
		long bitFlag(int a) { return 1L << (long)a; }
		boolean isFlagged(long x, int a) { return (x & bitFlag(a)) != 0; }

		long countString(String str, String a) { return (str.length() - str.replace(a, "").length()) / a.length(); }
		long countStringAll(String str, String a) { return str.length() - str.replaceAll(a, "").length(); }


		String reverse(String str) { return (new StringBuilder(str)).reverse().toString(); }
		void reverse(String[] array) { for(int i = 0; i < array.length / 2; i ++) swap(array, i, array.length - i - 1); }
		void reverse(int[] array) { for(int i = 0; i < array.length / 2; i ++) swap(array, i, array.length - i - 1); }
		void reverse(long[] array) { for(int i = 0; i < array.length / 2; i ++) swap(array, i, array.length - i - 1); }
		void reverse(double[] array) { for(int i = 0; i < array.length / 2; i ++) swap(array, i, array.length - i - 1); }
		void reverse(char[] array) { for(int i = 0; i < array.length / 2; i ++) swap(array, i, array.length - i - 1); }
		void reverse(boolean[] array) { for(int i = 0; i < array.length / 2; i ++) swap(array, i, array.length - i - 1); }
		<T> void reverse(T[] array) { for(int i = 0; i < array.length / 2; i ++) swap(array, i, array.length - i - 1); }
		void fill(int[] array, int x) { Arrays.fill(array, x); }
		void fill(long[] array, long x) { Arrays.fill(array, x); }
		void fill(double[] array, double x) { Arrays.fill(array, x); }
		void fill(char[] array, char x) { Arrays.fill(array, x); }
		void fill(boolean[] array, boolean x) { Arrays.fill(array, x); }
		void fill(int[][] array, int x) { for(int[] a : array) fill(a, x); }
		void fill(long[][] array, long x) { for(long[] a : array) fill(a, x); }
		void fill(double[][] array, double x) { for(double[] a : array) fill(a, x); }
		void fill(char[][] array, char x) { for(char[] a : array) fill(a, x); }
		void fill(boolean[][] array, boolean x) { for(boolean[] a : array) fill(a, x); }
		void fill(int[][][] array, int x) { for(int[][] a : array) fill(a, x); }
		void fill(long[][][] array, long x) { for(long[][] a : array) fill(a, x); }
		void fill(double[][][] array, double x) { for(double[][] a : array) fill(a, x); }
		void fill(char[][][] array, char x) { for(char[][] a : array) fill(a, x); }
		void fill(boolean[][][] array, boolean x) { for(boolean[][] a : array) fill(a, x); }

		int[] resize(int[] array, int m, int x) {
			int resized[] = new int[m];
			for(int i = max(0, - x); i < array.length && i + x < m; i ++) resized[i + x] = array[i];
			return resized;
		}
		long[] resize(long[] array, int m, int x) {
			long resized[] = new long[m];
			for(int i = max(0, - x); i < array.length && i + x < m; i ++) resized[i + x] = array[i];
			return resized;
		}
		double[] resize(double[] array, int m, int x) {
			double resized[] = new double[m];
			for(int i = max(0, - x); i < array.length && i + x < m; i ++) resized[i + x] = array[i];
			return resized;
		}
		char[] resize(char[] array, int m, int x) {
			char resized[] = new char[m];
			for(int i = max(0, - x); i < array.length && i + x < m; i ++) resized[i + x] = array[i];
			return resized;
		}
		boolean[] resize(boolean[] array, int m, int x) {
			boolean resized[] = new boolean[m];
			for(int i = max(0, - x); i < array.length && i + x < m; i ++) resized[i + x] = array[i];
			return resized;
		}
		Object[] resize(Object[] array, int m, int x) {
			Object resized[] = new Object[m];
			for(int i = max(0, - x); i < array.length && i + x < m; i ++) resized[i + x] = array[i];
			return resized;
		}

		void shuffleArray(int[] array){
			for(int i = 0; i < array.length; i ++){
				int tmp = array[i];
				int randomPos = random(i, array.length);
				array[i] = array[randomPos];
				array[randomPos] = tmp;
			}
		}
		void shuffleArray(long[] array){
			for(int i = 0; i < array.length; i ++){
				long tmp = array[i];
				int randomPos = random(i, array.length);
				array[i] = array[randomPos];
				array[randomPos] = tmp;
			}
		}
		void shuffleArray(double[] array){
			for(int i = 0; i < array.length; i ++){
				double tmp = array[i];
				int randomPos = random(i, array.length);
				array[i] = array[randomPos];
				array[randomPos] = tmp;
			}
		}
		int[] randomi(int num, int max){
			int array[] = new int[num];
			for(int i = 0; i < num; i ++) array[i] = random(max);
			return array;
		}
		long[] randoml(int num, long max){
			long array[] = new long[num];
			for(int i = 0; i < num; i ++) array[i] = random(max);
			return array;
		}
		double[] randomd(int num, double max){
			double array[] = new double[num];
			for(int i = 0; i < num; i ++) array[i] = random(max);
			return array;
		}
		int[] randomi(int num, int min, int max){
			int array[] = new int[num];
			for(int i = 0; i < num; i ++) array[i] = random(min, max);
			return array;
		}
		long[] randoml(int num, long min, long max){
			long array[] = new long[num];
			for(int i = 0; i < num; i ++) array[i] = random(min, max);
			return array;
		}
		double[] randomd(int num, double min, double max){
			double array[] = new double[num];
			for(int i = 0; i < num; i ++) array[i] = random(min, max);
			return array;
		}

		void swap(String[] array, int i, int j) {
			String tmp = array[i];
			array[i] = array[j];
			array[j] = tmp;
		}
		void swap(int[] array, int i, int j) {
			int tmp = array[i];
			array[i] = array[j];
			array[j] = tmp;
		}
		void swap(long[] array, int i, int j) {
			long tmp = array[i];
			array[i] = array[j];
			array[j] = tmp;
		}
		void swap(double[] array, int i, int j) {
			double tmp = array[i];
			array[i] = array[j];
			array[j] = tmp;
		}
		void swap(char[] array, int i, int j) {
			char tmp = array[i];
			array[i] = array[j];
			array[j] = tmp;
		}
		void swap(boolean[] array, int i, int j) {
			boolean tmp = array[i];
			array[i] = array[j];
			array[j] = tmp;
		}
		<T> void swap(T[] array, int i, int j) {
			T tmp = array[i];
			array[i] = array[j];
			array[j] = tmp;
		}

		int[] compress(int[] a) {
			int num = a.length;
			Set<Integer> ss = new TreeSet<>();
			for(int i = 0; i < num; i ++) ss.add(a[i]);
			int compressed[] = new int[ss.size()];
			int j = 0;
			for(Integer x : ss) compressed[j ++] = x;
			for(int i = 0; i < num; i ++) a[i] = lowerBound(compressed, a[i]);
			return compressed;
		}
		long[] compress(long[] a) {
			int num = a.length;
			Set<Long> ss = new TreeSet<>();
			for(int i = 0; i < num; i ++) ss.add(a[i]);
			long compressed[] = new long[ss.size()];
			int j = 0;
			for(Long x : ss) compressed[j ++] = x;
			for(int i = 0; i < num; i ++) a[i] = lowerBound(compressed, a[i]);
			return compressed;
		}
		double[] compress(double[] a) {
			int num = a.length;
			Set<Double> ss = new TreeSet<>();
			for(int i = 0; i < num; i ++) ss.add(a[i]);
			double compressed[] = new double[ss.size()];
			int j = 0;
			for(Double x : ss) compressed[j ++] = x;
			for(int i = 0; i < num; i ++) a[i] = lowerBound(compressed, a[i]);
			return compressed;
		}


		int lowerBound(int[] array, int key) {
			return BS(array, key, true, true, true);
		}
		int lowerBound(int[] array, int key, int ng, int ok) {
			return BS(array, key, true, true, true, ng, ok);
		}
		int upperBound(int[] array, int key) {
			return BS(array, key, true, true, false);
		}
		int upperBound(int[] array, int key, int ng, int ok) {
			return BS(array, key, true, true, false, ng, ok);
		}
		int cntBS(int[] array, int key, boolean ascending, boolean greater, boolean equals) {
			return BS(array, key, ascending, greater, equals, true);
		}
		int cntBS(int[] array, int key, boolean ascending, boolean greater, boolean equals, int ng, int ok) {
			return BS(array, key, ascending, greater, equals, true, ng, ok);
		}
		int BS(int[] array, int key, boolean ascending, boolean greater, boolean equals) {
			return BS(array, key, ascending, greater, equals, false);
		}
		int BS(int[] array, int key, boolean ascending, boolean greater, boolean equals, int ng, int ok) {
			return BS(array, key, ascending, greater, equals, false, ng, ok);
		}
		int BS(int[] array, int key, boolean ascending, boolean greater, boolean equals, boolean count) {
			int ng = ascending ^ greater ? array.length : -1;
			int ok = ascending ^ greater ? -1 : array.length;
			return BS(array, key, ascending, greater, equals, count, ng, ok);
		}
		int BS(int[] array, int key, boolean ascending, boolean greater, boolean equals, boolean count, int ng, int ok) {
			int index = binarySearch(array, key, greater, equals, ng, ok);
			return count ? (int)abs(ok - index) : index;
		}
		int binarySearch(int[] array, int key, boolean greater, boolean equals, int ng, int ok) {
			while (abs(ok - ng) > 1) {
				int mid = (ok + ng) / 2;
				if(isOKforBinarySearch(array, mid, key, greater, equals)) ok = mid; else ng = mid;
			}
			return ok;
		}
		boolean isOKforBinarySearch(int[] array, int index, int key, boolean greater, boolean equals) {
			return (array[index] > key && greater) || (array[index] < key && !greater) || (array[index] == key && equals);
		}
		int lowerBound(long[] array, long key) {
			return BS(array, key, true, true, true);
		}
		int lowerBound(long[] array, long key, int ng, int ok) {
			return BS(array, key, true, true, true, ng, ok);
		}
		int upperBound(long[] array, long key) {
			return BS(array, key, true, true, false);
		}
		int upperBound(long[] array, long key, int ng, int ok) {
			return BS(array, key, true, true, false, ng, ok);
		}
		int cntBS(long[] array, long key, boolean ascending, boolean greater, boolean equals) {
			return BS(array, key, ascending, greater, equals, true);
		}
		int cntBS(long[] array, long key, boolean ascending, boolean greater, boolean equals, int ng, int ok) {
			return BS(array, key, ascending, greater, equals, true, ng, ok);
		}
		int BS(long[] array, long key, boolean ascending, boolean greater, boolean equals) {
			return BS(array, key, ascending, greater, equals, false);
		}
		int BS(long[] array, long key, boolean ascending, boolean greater, boolean equals, int ng, int ok) {
			return BS(array, key, ascending, greater, equals, false, ng, ok);
		}
		int BS(long[] array, long key, boolean ascending, boolean greater, boolean equals, boolean count) {
			int ng = ascending ^ greater ? array.length : -1;
			int ok = ascending ^ greater ? -1 : array.length;
			return BS(array, key, ascending, greater, equals, count, ng, ok);
		}
		int BS(long[] array, long key, boolean ascending, boolean greater, boolean equals, boolean count, int ng, int ok) {
			int index = binarySearch(array, key, greater, equals, ng, ok);
			return count ? (int)abs(ok - index) : index;
		}
		int binarySearch(long[] array, long key, boolean greater, boolean equals, int ng, int ok) {
			while (abs(ok - ng) > 1) {
				int mid = (ok + ng) / 2;
				if(isOKforBinarySearch(array, mid, key, greater, equals)) ok = mid; else ng = mid;
			}
			return ok;
		}
		boolean isOKforBinarySearch(long[] array, int index, long key, boolean greater, boolean equals) {
			return (array[index] > key && greater) || (array[index] < key && !greater) || (array[index] == key && equals);
		}
		int lowerBound(double[] array, double key) {
			return BS(array, key, true, true, true);
		}
		int lowerBound(double[] array, double key, int ng, int ok) {
			return BS(array, key, true, true, true, ng, ok);
		}
		int upperBound(double[] array, double key) {
			return BS(array, key, true, true, false);
		}
		int upperBound(double[] array, double key, int ng, int ok) {
			return BS(array, key, true, true, false, ng, ok);
		}
		int cntBS(double[] array, double key, boolean ascending, boolean greater, boolean equals) {
			return BS(array, key, ascending, greater, equals, true);
		}
		int cntBS(double[] array, double key, boolean ascending, boolean greater, boolean equals, int ng, int ok) {
			return BS(array, key, ascending, greater, equals, true, ng, ok);
		}
		int BS(double[] array, double key, boolean ascending, boolean greater, boolean equals) {
			return BS(array, key, ascending, greater, equals, false);
		}
		int BS(double[] array, double key, boolean ascending, boolean greater, boolean equals, int ng, int ok) {
			return BS(array, key, ascending, greater, equals, false, ng, ok);
		}
		int BS(double[] array, double key, boolean ascending, boolean greater, boolean equals, boolean count) {
			int ng = ascending ^ greater ? array.length : -1;
			int ok = ascending ^ greater ? -1 : array.length;
			return BS(array, key, ascending, greater, equals, count, ng, ok);
		}
		int BS(double[] array, double key, boolean ascending, boolean greater, boolean equals, boolean count, int ng, int ok) {
			int index = binarySearch(array, key, greater, equals, ng, ok);
			return count ? (int)abs(ok - index) : index;
		}
		int binarySearch(double[] array, double key, boolean greater, boolean equals, int ng, int ok) {
			while (abs(ok - ng) > 1) {
				int mid = (ok + ng) / 2;
				if(isOKforBinarySearch(array, mid, key, greater, equals)) ok = mid; else ng = mid;
			}
			return ok;
		}
		boolean isOKforBinarySearch(double[] array, int index, double key, boolean greater, boolean equals) {
			return (array[index] > key && greater) || (array[index] < key && !greater) || (array[index] == key && equals);
		}
		<T extends Comparable<? super T>> int lowerBound(T[] array, T key) {
			return BS(array, key, true, true, true);
		}
		<T extends Comparable<? super T>> int lowerBound(T[] array, T key, int ng, int ok) {
			return BS(array, key, true, true, true, ng, ok);
		}
		<T extends Comparable<? super T>> int upperBound(T[] array, T key) {
			return BS(array, key, true, true, false);
		}
		<T extends Comparable<? super T>> int upperBound(T[] array, T key, int ng, int ok) {
			return BS(array, key, true, true, false, ng, ok);
		}
		<T extends Comparable<? super T>> int cntBS(T[] array, T key, boolean ascending, boolean greater, boolean equals) {
			return BS(array, key, ascending, greater, equals, true);
		}
		<T extends Comparable<? super T>> int cntBS(T[] array, T key, boolean ascending, boolean greater, boolean equals, int ng, int ok) {
			return BS(array, key, ascending, greater, equals, true, ng, ok);
		}
		<T extends Comparable<? super T>> int BS(T[] array, T key, boolean ascending, boolean greater, boolean equals) {
			return BS(array, key, ascending, greater, equals, false);
		}
		<T extends Comparable<? super T>> int BS(T[] array, T key, boolean ascending, boolean greater, boolean equals, int ng, int ok) {
			return BS(array, key, ascending, greater, equals, false, ng, ok);
		}
		<T extends Comparable<? super T>> int BS(T[] array, T key, boolean ascending, boolean greater, boolean equals, boolean count) {
			int ng = ascending ^ greater ? array.length : -1;
			int ok = ascending ^ greater ? -1 : array.length;
			return BS(array, key, ascending, greater, equals, count, ng, ok);
		}
		<T extends Comparable<? super T>> int BS(T[] array, T key, boolean ascending, boolean greater, boolean equals, boolean count, int ng, int ok) {
			int index = binarySearch(array, key, greater, equals, ng, ok);
			return count ? (int)abs(ok - index) : index;
		}
		<T extends Comparable<? super T>> int binarySearch(T[] array, T key, boolean greater, boolean equals, int ng, int ok) {
			while (abs(ok - ng) > 1) {
				int mid = (ok + ng) / 2;
				if(isOKforBinarySearch(array, mid, key, greater, equals)) ok = mid; else ng = mid;
			}
			return ok;
		}
		<T extends Comparable<? super T>> boolean isOKforBinarySearch(T[] array, int index, T key, boolean greater, boolean equals) {
			int compare = array[index].compareTo(key);
			return (compare > 0 && greater) || (compare < 0 && !greater) || (compare == 0 && equals);
		}
		<T> int lowerBound(T[] array, T key, Comparator<? super T> c) {
			return BS(array, key, true, true, true, c);
		}
		<T> int lowerBound(T[] array, T key, int ng, int ok, Comparator<? super T> c) {
			return BS(array, key, true, true, true, ng, ok, c);
		}
		<T> int upperBound(T[] array, T key, Comparator<? super T> c) {
			return BS(array, key, true, true, false, c);
		}
		<T> int upperBound(T[] array, T key, int ng, int ok, Comparator<? super T> c) {
			return BS(array, key, true, true, false, ng, ok, c);
		}
		<T> int cntBS(T[] array, T key, boolean ascending, boolean greater, boolean equals, Comparator<? super T> c) {
			return BS(array, key, ascending, greater, equals, true, c);
		}
		<T> int cntBS(T[] array, T key, boolean ascending, boolean greater, boolean equals, int ng, int ok, Comparator<? super T> c) {
			return BS(array, key, ascending, greater, equals, true, ng, ok, c);
		}
		<T> int BS(T[] array, T key, boolean ascending, boolean greater, boolean equals, Comparator<? super T> c) {
			return BS(array, key, ascending, greater, equals, false, c);
		}
		<T> int BS(T[] array, T key, boolean ascending, boolean greater, boolean equals, int ng, int ok, Comparator<? super T> c) {
			return BS(array, key, ascending, greater, equals, false, ng, ok, c);
		}
		<T> int BS(T[] array, T key, boolean ascending, boolean greater, boolean equals, boolean count, Comparator<? super T> c) {
			int ng = ascending ^ greater ? array.length : -1;
			int ok = ascending ^ greater ? -1 : array.length;
			return BS(array, key, ascending, greater, equals, count, ng, ok, c);
		}
		<T> int BS(T[] array, T key, boolean ascending, boolean greater, boolean equals, boolean count, int ng, int ok, Comparator<? super T> c) {
			int index = binarySearch(array, key, greater, equals, ng, ok, c);
			return count ? (int)abs(ok - index) : index;
		}
		<T> int binarySearch(T[] array, T key, boolean greater, boolean equals, int ng, int ok, Comparator<? super T> c) {
			while (abs(ok - ng) > 1) {
				int mid = (ok + ng) / 2;
				if(isOKforBinarySearch(array, mid, key, greater, equals, c)) ok = mid; else ng = mid;
			}
			return ok;
		}
		<T> boolean isOKforBinarySearch(T[] array, int index, T key, boolean greater, boolean equals, Comparator<? super T> c) {
			int compare = c.compare(array[index], key);
			return (compare > 0 && greater) || (compare < 0 && !greater) || (compare == 0 && equals);
		}
		<T extends Comparable<? super T>> int lowerBound(List<T> array, T key) {
			return BS(array, key, true, true, true);
		}
		<T extends Comparable<? super T>> int lowerBound(List<T> array, T key, int ng, int ok) {
			return BS(array, key, true, true, true, ng, ok);
		}
		<T extends Comparable<? super T>> int upperBound(List<T> array, T key) {
			return BS(array, key, true, true, false);
		}
		<T extends Comparable<? super T>> int upperBound(List<T> array, T key, int ng, int ok) {
			return BS(array, key, true, true, false, ng, ok);
		}
		<T extends Comparable<? super T>> int cntBS(List<T> array, T key, boolean ascending, boolean greater, boolean equals) {
			return BS(array, key, ascending, greater, equals, true);
		}
		<T extends Comparable<? super T>> int cntBS(List<T> array, T key, boolean ascending, boolean greater, boolean equals, int ng, int ok) {
			return BS(array, key, ascending, greater, equals, true, ng, ok);
		}
		<T extends Comparable<? super T>> int BS(List<T> array, T key, boolean ascending, boolean greater, boolean equals) {
			return BS(array, key, ascending, greater, equals, false);
		}
		<T extends Comparable<? super T>> int BS(List<T> array, T key, boolean ascending, boolean greater, boolean equals, int ng, int ok) {
			return BS(array, key, ascending, greater, equals, false, ng, ok);
		}
		<T extends Comparable<? super T>> int BS(List<T> array, T key, boolean ascending, boolean greater, boolean equals, boolean count) {
			int ng = ascending ^ greater ? array.size() : -1;
			int ok = ascending ^ greater ? -1 : array.size();
			return BS(array, key, ascending, greater, equals, count, ng, ok);
		}
		<T extends Comparable<? super T>> int BS(List<T> array, T key, boolean ascending, boolean greater, boolean equals, boolean count, int ng, int ok) {
			int index = binarySearch(array, key, greater, equals, ng, ok);
			return count ? (int)abs(ok - index) : index;
		}
		<T extends Comparable<? super T>> int binarySearch(List<T> array, T key, boolean greater, boolean equals, int ng, int ok) {
			while (abs(ok - ng) > 1) {
				int mid = (ok + ng) / 2;
				if(isOKforBinarySearch(array, mid, key, greater, equals)) ok = mid; else ng = mid;
			}
			return ok;
		}
		<T extends Comparable<? super T>> boolean isOKforBinarySearch(List<T> array, int index, T key, boolean greater, boolean equals) {
			int compare = array.get(index).compareTo(key);
			return (compare > 0 && greater) || (compare < 0 && !greater) || (compare == 0 && equals);
		}
		<T> int lowerBound(List<T> array, T key, Comparator<? super T> c) {
			return BS(array, key, true, true, true, c);
		}
		<T> int lowerBound(List<T> array, T key, int ng, int ok, Comparator<? super T> c) {
			return BS(array, key, true, true, true, ng, ok, c);
		}
		<T> int upperBound(List<T> array, T key, Comparator<? super T> c) {
			return BS(array, key, true, true, false, c);
		}
		<T> int upperBound(List<T> array, T key, int ng, int ok, Comparator<? super T> c) {
			return BS(array, key, true, true, false, ng, ok, c);
		}
		<T> int cntBS(List<T> array, T key, boolean ascending, boolean greater, boolean equals, Comparator<? super T> c) {
			return BS(array, key, ascending, greater, equals, true, c);
		}
		<T> int cntBS(List<T> array, T key, boolean ascending, boolean greater, boolean equals, int ng, int ok, Comparator<? super T> c) {
			return BS(array, key, ascending, greater, equals, true, ng, ok, c);
		}
		<T> int BS(List<T> array, T key, boolean ascending, boolean greater, boolean equals, Comparator<? super T> c) {
			return BS(array, key, ascending, greater, equals, false, c);
		}
		<T> int BS(List<T> array, T key, boolean ascending, boolean greater, boolean equals, int ng, int ok, Comparator<? super T> c) {
			return BS(array, key, ascending, greater, equals, false, ng, ok, c);
		}
		<T> int BS(List<T> array, T key, boolean ascending, boolean greater, boolean equals, boolean count, Comparator<? super T> c) {
			int ng = ascending ^ greater ? array.size() : -1;
			int ok = ascending ^ greater ? -1 : array.size();
			return BS(array, key, ascending, greater, equals, count, ng, ok, c);
		}
		<T> int BS(List<T> array, T key, boolean ascending, boolean greater, boolean equals, boolean count, int ng, int ok, Comparator<? super T> c) {
			int index = binarySearch(array, key, greater, equals, ng, ok, c);
			return count ? (int)abs(ok - index) : index;
		}
		<T> int binarySearch(List<T> array, T key, boolean greater, boolean equals, int ng, int ok, Comparator<? super T> c) {
			while (abs(ok - ng) > 1) {
				int mid = (ok + ng) / 2;
				if(isOKforBinarySearch(array, mid, key, greater, equals, c)) ok = mid; else ng = mid;
			}
			return ok;
		}
		<T> boolean isOKforBinarySearch(List<T> array, int index, T key, boolean greater, boolean equals, Comparator<? super T> c) {
			int compare = c.compare(array.get(index), key);
			return (compare > 0 && greater) || (compare < 0 && !greater) || (compare == 0 && equals);
		}

		PairLL binaryRangeSearch(long left, long right, UnaryOperator<Long> op, boolean minimize) {
			long ok1 = right, ng1 = left;
			while(abs(ok1 - ng1) > 1) {
				long mid = (ok1 + ng1) / 2;
				boolean isOK = (op.apply(mid + 1) - op.apply(mid)) * (minimize ? 1 : -1) >= 0;
				if(isOK) ok1 = mid; else ng1 = mid;
			}
			long ok2 = left, ng2 = right;
			while(abs(ok2 - ng2) > 1) {
				long mid = (ok2 + ng2) / 2;
				boolean isOK = (op.apply(mid - 1) - op.apply(mid)) * (minimize ? 1 : -1) >= 0;
				if(isOK) ok2 = mid; else ng2 = mid;
			}
			return new PairLL(ok1, ok2); //[l, r]
		}

		double ternarySearch(double left, double right, UnaryOperator<Double> op, boolean minimize, int loop) {
			for(int cnt = 0; cnt < loop; cnt ++) {
				double m1 = (left * 2 + right) / 3.0;
				double m2 = (left + right * 2) / 3.0;
				if(op.apply(m1) > op.apply(m2) ^ minimize) right = m2; else left = m1;
			}
			return (left + right) / 2.0;
		}


		// mods
		final long MOD = (long)1e9 + 7; // 998244353;
		long mod(long i) { i %= MOD; return i + (i < 0 ? MOD : 0); }
		void mod(long[] a) { for(int i = 0; i < a.length; i ++) a[i] = mod(a[i]); }
		void mod(long[][] a) { for(int i = 0; i < a.length; i ++) mod(a[i]); }
		void mod(long[][][] a) { for(int i = 0; i < a.length; i ++) mod(a[i]); }

		long pow_m(long x, long y) {
			long ans = 1;
			for(; y > 0; y /= 2) {
				if(y % 2 != 0) ans = mod(ans * x);
				x = mod(x * x);
			}
			return ans;
		}
		long[] pows_m(long x, int max) {
			long pow[] = new long[max + 1];
			pow[0] = 1;
			for(int i = 0; i < max; i ++) pow[i + 1] = mod(pow[i] * x);
			return pow;
		}
		long fact_m(int n) {
			long ans = 1;
			for(int i = 1; i <= n; i ++) ans = mod(ans * i);
			return ans;
		}

		final int MAX_INV_SIZE = 100_100;
		Map<Long, Long> invMap = new HashMap<>();
		long inv(long x) {
			x = mod(x);
			if(invMap.containsKey(x)) return invMap.get(x);
			if(invMap.size() >= MAX_INV_SIZE) return calInv(x);
			invMap.put(x, calInv(x));
			return invMap.get(x);
		}
		long calInv(long x) { return pow_m(x, MOD - 2); }

		final int MAX_FACT = 5_000_100;
		long fact[];
		long invFact[];
		boolean isFactPrepared = false;
		Map<Integer, long[]> factMap;
		void prepareFact() {
			fact = new long[MAX_FACT];
			fill(fact, 0);
			invFact = new long[MAX_FACT];
			fill(invFact, 0);
			fact[0] = 1;
			int maxIndex = min(MAX_FACT, (int)MOD);
			for(int i = 1; i < maxIndex; i ++) fact[i] = mod(fact[i - 1] * i);
			invFact[maxIndex - 1] = inv(fact[maxIndex - 1]);
			for(int i = maxIndex - 1; i > 0; i --) invFact[i - 1] = mod(invFact[i] * i);

			factMap = new HashMap<>();
			isFactPrepared = true;
		}

		long P(int n, int r) {
			if(!isFactPrepared) { prepareFact(); }
			if(n < 0 || r < 0 || n < r) { return 0; }
			if(n >= MAX_FACT) {
				if(!factMap.containsKey(n)) {
					long largeFact[] = new long[MAX_FACT];
					factMap.put(n, largeFact);
					fill(largeFact, -INF);
					largeFact[0] = 1;
				}
				long largeFact[] = factMap.get(n);
				int i = r;
				while(isINF(largeFact[i])) i --;
				for(; i < r; i ++) largeFact[i + 1] = mod(largeFact[i] * (n - i));
				return largeFact[r];
			}
			return mod(fact[n] * invFact[n - r]);
		}
		long C(int n, int r) {
			if(!isFactPrepared) prepareFact();
			if(n < 0 || r < 0 || n < r) return 0;
			return mod(P(n, r) * invFact[r]);
		}
		long H(int n, int r) { return C((n - 1) + r, r); }


		// grid
		class Grids {
			int h;
			int w;
			Grid[][] gs;
			Grid[] gi;
			Grids(int h, int w) {
				this.h = h;
				this.w = w;
				gs = new Grid[h][w];
				gi = new Grid[h * w];
				for(int i = 0; i < h; i ++) {
					for(int j = 0; j < w; j ++) {
						gs[i][j] = new Grid(i, j, h, w);
						gi[gs[i][j].i] = gs[i][j];
					}
				}
			}

			void init(boolean[][] b) {
				for(int i = 0; i < h; i ++) for(int j = 0; j < w; j ++) gs[i][j].b = b[i][j];
			}
			void init(long[][] val) {
				for(int i = 0; i < h; i ++) for(int j = 0; j < w; j ++) gs[i][j].val = val[i][j];
			}

			Grid get(int x, int y) { return isValid(x, y, h, w) ? gs[x][y] : null; }
			Grid get(int i) { return get(i / w, i % w); }

			int dx[] = {0, -1, 1, 0, 0, -1, 1, -1, 1};
			int dy[] = {0, 0, 0, -1, 1, -1, -1, 1, 1};
			Grid next(int x, int y, int i) { return next(gs[x][y], i); }
			Grid next(Grid g, int i) {
				return isValid(g.x + dx[i], g.y + dy[i], g.h, g.w)
					? gs[g.x + dx[i]][g.y + dy[i]]
					: null;
			}
		}
		class Grid implements Comparable<Grid> {
			int x;
			int y;
			int h;
			int w;
			int i;
			boolean b;
			long val;

			Grid() {  }
			Grid(int x, int y, int h, int w) { init(x, y, h, w, false, 0); }
			Grid(int x, int y, int h, int w, boolean b) { init(x, y, h, w, b, 0); }
			Grid(int x, int y, int h, int w, long val) { init(x, y, h, w, false, val); }
			Grid(int x, int y, int h, int w, boolean b, long val) { init(x, y, h, w, b, val); }

			void init(int x, int y, int h, int w, boolean b, long val) {
				this.x = x;
				this.y = y;
				this.h = h;
				this.w = w;
				this.b = b;
				this.val = val;
				this.i = x * w + y;
			}

			@Override
			public String toString() { return "("+x+", "+y+")"+" "+booleanToString(b)+" "+val; }
			@Override
			public int hashCode() { return Objects.hash(x, y, h, w, b, val); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null) return false;
				if(this.getClass() != obj.getClass()) return false;
				Grid that = (Grid) obj;
				if(this.x != that.x) return false;
				if(this.y != that.y) return false;
				if(this.h != that.h) return false;
				if(this.w != that.w) return false;
				if(this.b != that.b) return false;
				if(this.val != that.val) return false;
				return true;
			}
			@Override
			public int compareTo(Grid that) {
				int c = Long.compare(this.val, that.val);
				if(c == 0) c = Integer.compare(this.x, that.x);
				if(c == 0) c = Integer.compare(this.y, that.y);
				return c;
			}
		}

		boolean isValid(int x, int y, int h, int w) { return x >= 0 && x < h && y >= 0 && y < w; }
		boolean isValid(Grid g) { return isValid(g.x, g.y, g.h, g.w); }

		// graph
		class Graph {
			int numNode;
			int numEdge;
			boolean directed;
			Set<Edge> edges = new HashSet<>();
			Node nodes[];
			Node reversedNodes[];

			Graph(int numNode, int numEdge, boolean directed) {
				this.numNode = numNode;
				this.numEdge = numEdge;
				this.directed = directed;
				nodes = new Node[numNode];
				reversedNodes = new Node[numNode];
				for(int i = 0; i < numNode; i ++) {
					nodes[i] = new Node(i);
					reversedNodes[i] = new Node(i);
				}
			}

			void init(Set<Edge> edges) {
				this.edges = edges;
				for(Edge e : edges) add(e);
			}

			void add(Edge e) {
				edges.add(e);
				nodes[e.source].add(e.target, e.cost);
				if(directed) reversedNodes[e.target].add(e.source, e.cost);
				else nodes[e.target].add(e.source, e.cost);
			}

			void remove(Edge e) {
				edges.remove(e);
				nodes[e.source].remove(e.target, e.cost);
				if(directed) reversedNodes[e.target].remove(e.source, e.cost);
				else nodes[e.target].remove(e.source, e.cost);
			}

			void update(Edge e, long cost) {
				nodes[e.source].update(e.target, cost);
				if(directed) reversedNodes[e.target].update(e.source, cost);
				else nodes[e.target].update(e.source, cost);
				e.cost = cost;
			}
			void update(int source, int target, long cost) { update(new Edge(source, target, cost), cost); }

			void clearNodes() {
				for(Node n : nodes) n.clear();
				for(Node n : reversedNodes) n.clear();
			}
		}

		class Node extends HashSet<Edge> {
			int id;

			Node(int id) { this.id = id; }
			void add(int target, long cost) { add(new Edge(id, target, cost)); }
			void remove(int target, long cost) { remove(new Edge(id, target, cost)); }
			void update(int target, long cost) { remove(target, cost); add(target, cost); }
		}

		class Edge implements Comparable<Edge> {
			int source;
			int target;
			long cost;
			Edge(int source, int target, long cost) {
				this.source = source;
				this.target = target;
				this.cost = cost;
			}


			@Override
			public String toString() { return source+" - "+cost+" -> "+target; }
			@Override
			public int hashCode() { return Objects.hash(source, target); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null) return false;
				if(this.getClass() != obj.getClass()) return false;
				Edge that = (Edge) obj;
				if(this.source != that.source) return false;
				if(this.target != that.target) return false;
				return true;
			}
			@Override
			public int compareTo(Edge that) {
				int c = Long.compare(this.cost, that.cost);
				if(c == 0) c = Integer.compare(this.source, that.source);
				if(c == 0) c = Integer.compare(this.target, that.target);
				return c;
			}
		}

		// Pair, Tuple
		class Pair<T extends Comparable<? super T>, U extends Comparable<? super U>> implements Comparable<Pair<T, U>> {
			T a;
			U b;
			Pair() { }
			Pair(T a, U b) {
				this.a = a;
				this.b = b;
			}

			@Override
			public String toString() { return "("+a.toString()+", "+b.toString()+")"; }
			@Override
			public int hashCode() { return Objects.hash(a, b); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null) return false;
				if(this.getClass() != obj.getClass()) return false;
				Pair that = (Pair) obj;
				if(this.a.getClass() != that.a.getClass()) return false;
				if(this.b.getClass() != that.b.getClass()) return false;
				if(!this.a.equals(that.a)) return false;
				if(!this.b.equals(that.b)) return false;
				return true;
			}
			@Override
			public int compareTo(Pair<T, U> that) {
				int c = (this.a).compareTo(that.a);
				if(c == 0) c = (this.b).compareTo(that.b);
				return c;
			}
		}
		class PairII implements Comparable<PairII> {
			int a; int b;
			PairII() { }
			PairII(int a, int b) { this.a = a; this.b = b; }
			@Override public String toString() { return "("+a+", "+b+")"; }
			@Override public int hashCode() { return Objects.hash(a, b); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null) return false;
				if(this.getClass() != obj.getClass()) return false;
				PairII that = (PairII) obj;
				if(this.a != that.a || this.b != that.b) return false;
				return true;
			}
			@Override
			public int compareTo(PairII that) {
				int c = Integer.compare(this.a, that.a);
				if(c == 0) c = Integer.compare(this.b, that.b);
				return c;
			}
		}
		class PairIL implements Comparable<PairIL> {
			int a; long b;
			PairIL() { }
			PairIL(int a, long b) { this.a = a; this.b = b; }
			@Override public String toString() { return "("+a+", "+b+")"; }
			@Override public int hashCode() { return Objects.hash(a, b); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null) return false;
				if(this.getClass() != obj.getClass()) return false;
				PairIL that = (PairIL) obj;
				if(this.a != that.a || this.b != that.b) return false;
				return true;
			}
			@Override
			public int compareTo(PairIL that) {
				int c = Integer.compare(this.a, that.a);
				if(c == 0) c = Long.compare(this.b, that.b);
				return c;
			}
		}
		class PairID implements Comparable<PairID> {
			int a; double b;
			PairID() { }
			PairID(int a, double b) { this.a = a; this.b = b; }
			@Override public String toString() { return "("+a+", "+b+")"; }
			@Override public int hashCode() { return Objects.hash(a, b); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null) return false;
				if(this.getClass() != obj.getClass()) return false;
				PairID that = (PairID) obj;
				if(this.a != that.a || this.b != that.b) return false;
				return true;
			}
			@Override
			public int compareTo(PairID that) {
				int c = Integer.compare(this.a, that.a);
				if(c == 0) c = Double.compare(this.b, that.b);
				return c;
			}
		}
		class PairLI implements Comparable<PairLI> {
			long a; int b;
			PairLI() { }
			PairLI(long a, int b) { this.a = a; this.b = b; }
			@Override public String toString() { return "("+a+", "+b+")"; }
			@Override public int hashCode() { return Objects.hash(a, b); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null) return false;
				if(this.getClass() != obj.getClass()) return false;
				PairLI that = (PairLI) obj;
				if(this.a != that.a || this.b != that.b) return false;
				return true;
			}
			@Override
			public int compareTo(PairLI that) {
				int c = Long.compare(this.a, that.a);
				if(c == 0) c = Integer.compare(this.b, that.b);
				return c;
			}
		}
		class PairLL implements Comparable<PairLL> {
			long a; long b;
			PairLL() { }
			PairLL(long a, long b) { this.a = a; this.b = b; }
			@Override public String toString() { return "("+a+", "+b+")"; }
			@Override public int hashCode() { return Objects.hash(a, b); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null) return false;
				if(this.getClass() != obj.getClass()) return false;
				PairLL that = (PairLL) obj;
				if(this.a != that.a || this.b != that.b) return false;
				return true;
			}
			@Override
			public int compareTo(PairLL that) {
				int c = Long.compare(this.a, that.a);
				if(c == 0) c = Long.compare(this.b, that.b);
				return c;
			}
		}
		class PairLD implements Comparable<PairLD> {
			long a; double b;
			PairLD() { }
			PairLD(long a, double b) { this.a = a; this.b = b; }
			@Override public String toString() { return "("+a+", "+b+")"; }
			@Override public int hashCode() { return Objects.hash(a, b); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null) return false;
				if(this.getClass() != obj.getClass()) return false;
				PairLD that = (PairLD) obj;
				if(this.a != that.a || this.b != that.b) return false;
				return true;
			}
			@Override
			public int compareTo(PairLD that) {
				int c = Long.compare(this.a, that.a);
				if(c == 0) c = Double.compare(this.b, that.b);
				return c;
			}
		}
		class PairDI implements Comparable<PairDI> {
			double a; int b;
			PairDI() { }
			PairDI(double a, int b) { this.a = a; this.b = b; }
			@Override public String toString() { return "("+a+", "+b+")"; }
			@Override public int hashCode() { return Objects.hash(a, b); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null) return false;
				if(this.getClass() != obj.getClass()) return false;
				PairDI that = (PairDI) obj;
				if(this.a != that.a || this.b != that.b) return false;
				return true;
			}
			@Override
			public int compareTo(PairDI that) {
				int c = Double.compare(this.a, that.a);
				if(c == 0) c = Integer.compare(this.b, that.b);
				return c;
			}
		}
		class PairDL implements Comparable<PairDL> {
			double a; long b;
			PairDL() { }
			PairDL(double a, long b) { this.a = a; this.b = b; }
			@Override public String toString() { return "("+a+", "+b+")"; }
			@Override public int hashCode() { return Objects.hash(a, b); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null) return false;
				if(this.getClass() != obj.getClass()) return false;
				PairDL that = (PairDL) obj;
				if(this.a != that.a || this.b != that.b) return false;
				return true;
			}
			@Override
			public int compareTo(PairDL that) {
				int c = Double.compare(this.a, that.a);
				if(c == 0) c = Long.compare(this.b, that.b);
				return c;
			}
		}
		class PairDD implements Comparable<PairDD> {
			double a; double b;
			PairDD() { }
			PairDD(double a, double b) { this.a = a; this.b = b; }
			@Override public String toString() { return "("+a+", "+b+")"; }
			@Override public int hashCode() { return Objects.hash(a, b); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null) return false;
				if(this.getClass() != obj.getClass()) return false;
				PairDD that = (PairDD) obj;
				if(this.a != that.a || this.b != that.b) return false;
				return true;
			}
			@Override
			public int compareTo(PairDD that) {
				int c = Double.compare(this.a, that.a);
				if(c == 0) c = Double.compare(this.b, that.b);
				return c;
			}
		}

		interface ITuple {
			public StringBuilder toStringBuilder();
			@Override
			public String toString();
			@Override
			public int hashCode();
			@Override
			public boolean equals(Object obj);
		}
		class BasicTuple<T extends ITuple & Comparable<? super T>, V extends Comparable<? super V>> implements Comparable<BasicTuple> {
			T t;
			V a;
			BasicTuple() {  }

			StringBuilder sbTuple = new StringBuilder();
			public StringBuilder toStringBuilder() {
				sbTuple.setLength(0);
				return sbTuple.append(t.toStringBuilder()).append(", ").append(a);
			}
			@Override
			public String toString() { return "("+toStringBuilder().toString()+")"; }
			@Override
			public int hashCode() { return Objects.hash(t, a); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null) return false;
				if(this.getClass() != obj.getClass()) return false;
				BasicTuple that = (BasicTuple) obj;
				if(this.t.getClass() != that.t.getClass()) return false;
				if(this.a.getClass() != that.a.getClass()) return false;
				if(!this.t.equals(that.t)) return false;
				if(!this.a.equals(that.a)) return false;
				return true;
			}
			@Override
			@SuppressWarnings("unchecked")
			public int compareTo(BasicTuple that) {
				int c = (this.t).compareTo((T) (Object) that.t);
				if(c == 0) c = (this.a).compareTo((V) (Object) that.a);
				return c;
			}
		}
		class UniqueTuple<V extends Comparable<? super V>> implements ITuple, Comparable<UniqueTuple> {
			V a;
			UniqueTuple() {  }

			StringBuilder sbTuple = new StringBuilder();
			public StringBuilder toStringBuilder() {
				sbTuple.setLength(0);
				return sbTuple.append(a);
			}
			@Override
			public String toString() { return "("+toStringBuilder().toString()+")"; }
			@Override
			public int hashCode() { return Objects.hash(a); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null) return false;
				if(this.getClass() != obj.getClass()) return false;
				UniqueTuple that = (UniqueTuple) obj;
				if(this.a.getClass() != that.a.getClass()) return false;
				if(!this.a.equals(that.a)) return false;
				return true;
			}
			@Override
			@SuppressWarnings("unchecked")
			public int compareTo(UniqueTuple that) {
				return (this.a).compareTo((V) (Object) that.a);
			}
		}

		class Tuple1<T0 extends Comparable<? super T0>> extends UniqueTuple<T0> implements ITuple {
			Tuple1() { super(); }
			Tuple1(T0 a0) {
				super();
				this.a = a0;
			}
			T0 get0() { return a; }
			void set0(T0 x) { a = x; }
		}
		class Tuple2<
				T0 extends Comparable<? super T0>,
				T1 extends Comparable<? super T1>>
				extends BasicTuple<Tuple1<T0>, T1> implements ITuple {
			Tuple2() { super(); }
			@SuppressWarnings("unchecked")
			Tuple2(T0 a0, T1 a1) {
				super();
				this.t = new Tuple1(a0);
				this.a = a1;
			}
			T0 get0() { return t.get0(); }
			T1 get1() { return a; }
			void set0(T0 x) { t.set0(x); }
			void set1(T1 x) { a = x; }
		}
		class Tuple3<
				T0 extends Comparable<? super T0>,
				T1 extends Comparable<? super T1>,
				T2 extends Comparable<? super T2>>
				extends BasicTuple<Tuple2<T0, T1>, T2> implements ITuple {
			Tuple3() { super(); }
			@SuppressWarnings("unchecked")
			Tuple3(T0 a0, T1 a1, T2 a2) {
				super();
				this.t = new Tuple2(a0, a1);
				this.a = a2;
			}
			T0 get0() { return t.get0(); }
			T1 get1() { return t.get1(); }
			T2 get2() { return a; }
			void set0(T0 x) { t.set0(x); }
			void set1(T1 x) { t.set1(x); }
			void set2(T2 x) { a = x; }
		}
		class Tuple4<
				T0 extends Comparable<? super T0>,
				T1 extends Comparable<? super T1>,
				T2 extends Comparable<? super T2>,
				T3 extends Comparable<? super T3>>
				extends BasicTuple<Tuple3<T0, T1, T2>, T3> implements ITuple {
			Tuple4() { super(); }
			@SuppressWarnings("unchecked")
			Tuple4(T0 a0, T1 a1, T2 a2, T3 a3) {
				super();
				this.t = new Tuple3(a0, a1, a2);
				this.a = a3;
			}
			T0 get0() { return t.get0(); }
			T1 get1() { return t.get1(); }
			T2 get2() { return t.get2(); }
			T3 get3() { return a; }
			void set0(T0 x) { t.set0(x); }
			void set1(T1 x) { t.set1(x); }
			void set2(T2 x) { t.set2(x); }
			void set3(T3 x) { a = x; }
		}
		class Tuple5<
				T0 extends Comparable<? super T0>,
				T1 extends Comparable<? super T1>,
				T2 extends Comparable<? super T2>,
				T3 extends Comparable<? super T3>,
				T4 extends Comparable<? super T4>>
				extends BasicTuple<Tuple4<T0, T1, T2, T3>, T4> implements ITuple {
			Tuple5() { super(); }
			@SuppressWarnings("unchecked")
			Tuple5(T0 a0, T1 a1, T2 a2, T3 a3, T4 a4) {
				super();
				this.t = new Tuple4(a0, a1, a2, a3);
				this.a = a4;
			}
			T0 get0() { return t.get0(); }
			T1 get1() { return t.get1(); }
			T2 get2() { return t.get2(); }
			T3 get3() { return t.get3(); }
			T4 get4() { return a; }
			void set0(T0 x) { t.set0(x); }
			void set1(T1 x) { t.set1(x); }
			void set2(T2 x) { t.set2(x); }
			void set3(T3 x) { t.set3(x); }
			void set4(T4 x) { a = x; }
		}
		class Tuple6<
				T0 extends Comparable<? super T0>,
				T1 extends Comparable<? super T1>,
				T2 extends Comparable<? super T2>,
				T3 extends Comparable<? super T3>,
				T4 extends Comparable<? super T4>,
				T5 extends Comparable<? super T5>>
				extends BasicTuple<Tuple5<T0, T1, T2, T3, T4>, T5> implements ITuple {
			Tuple6() { super(); }
			@SuppressWarnings("unchecked")
			Tuple6(T0 a0, T1 a1, T2 a2, T3 a3, T4 a4, T5 a5) {
				super();
				this.t = new Tuple5(a0, a1, a2, a3, a4);
				this.a = a5;
			}
			T0 get0() { return t.get0(); }
			T1 get1() { return t.get1(); }
			T2 get2() { return t.get2(); }
			T3 get3() { return t.get3(); }
			T4 get4() { return t.get4(); }
			T5 get5() { return a; }
			void set0(T0 x) { t.set0(x); }
			void set1(T1 x) { t.set1(x); }
			void set2(T2 x) { t.set2(x); }
			void set3(T3 x) { t.set3(x); }
			void set4(T4 x) { t.set4(x); }
			void set5(T5 x) { a = x; }
		}
		class Tuple7<
				T0 extends Comparable<? super T0>,
				T1 extends Comparable<? super T1>,
				T2 extends Comparable<? super T2>,
				T3 extends Comparable<? super T3>,
				T4 extends Comparable<? super T4>,
				T5 extends Comparable<? super T5>,
				T6 extends Comparable<? super T6>>
				extends BasicTuple<Tuple6<T0, T1, T2, T3, T4, T5>, T6> implements ITuple {
			Tuple7() { super(); }
			@SuppressWarnings("unchecked")
			Tuple7(T0 a0, T1 a1, T2 a2, T3 a3, T4 a4, T5 a5, T6 a6) {
				super();
				this.t = new Tuple6(a0, a1, a2, a3, a4, a5);
				this.a = a6;
			}
			T0 get0() { return t.get0(); }
			T1 get1() { return t.get1(); }
			T2 get2() { return t.get2(); }
			T3 get3() { return t.get3(); }
			T4 get4() { return t.get4(); }
			T5 get5() { return t.get5(); }
			T6 get6() { return a; }
			void set0(T0 x) { t.set0(x); }
			void set1(T1 x) { t.set1(x); }
			void set2(T2 x) { t.set2(x); }
			void set3(T3 x) { t.set3(x); }
			void set4(T4 x) { t.set4(x); }
			void set5(T5 x) { t.set5(x); }
			void set6(T6 x) { a = x; }
		}
		class Tuple8<
				T0 extends Comparable<? super T0>,
				T1 extends Comparable<? super T1>,
				T2 extends Comparable<? super T2>,
				T3 extends Comparable<? super T3>,
				T4 extends Comparable<? super T4>,
				T5 extends Comparable<? super T5>,
				T6 extends Comparable<? super T6>,
				T7 extends Comparable<? super T7>>
				extends BasicTuple<Tuple7<T0, T1, T2, T3, T4, T5, T6>, T7> implements ITuple {
			Tuple8() { super(); }
			@SuppressWarnings("unchecked")
			Tuple8(T0 a0, T1 a1, T2 a2, T3 a3, T4 a4, T5 a5, T6 a6, T7 a7) {
				super();
				this.t = new Tuple7(a0, a1, a2, a3, a4, a5, a6);
				this.a = a7;
			}
			T0 get0() { return t.get0(); }
			T1 get1() { return t.get1(); }
			T2 get2() { return t.get2(); }
			T3 get3() { return t.get3(); }
			T4 get4() { return t.get4(); }
			T5 get5() { return t.get5(); }
			T6 get6() { return t.get6(); }
			T7 get7() { return a; }
			void set0(T0 x) { t.set0(x); }
			void set1(T1 x) { t.set1(x); }
			void set2(T2 x) { t.set2(x); }
			void set3(T3 x) { t.set3(x); }
			void set4(T4 x) { t.set4(x); }
			void set5(T5 x) { t.set5(x); }
			void set6(T6 x) { t.set6(x); }
			void set7(T7 x) { a = x; }
		}

		class TupleIII implements Comparable<TupleIII> {
			int a; int b; int c;
			TupleIII() {  }
			TupleIII(int a, int b, int c) { this.a = a; this.b = b; this.c = c; }
			@Override public String toString() { return "("+a+", "+b+", "+c+")"; }
			@Override public int hashCode() { return Objects.hash(a, b, c); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null || this.getClass() != obj.getClass()) return false;
				TupleIII that = (TupleIII) obj;
				if(this.a != that.a || this.b != that.b || this.c != that.c) return false;
				return true;
			}
			@Override
			public int compareTo(TupleIII that) {
				int c = Integer.compare(this.a, that.a);
				if(c == 0) c = Integer.compare(this.b, that.b);
				if(c == 0) c = Integer.compare(this.c, that.c);
				return c;
			}
		}
		class TupleIIL implements Comparable<TupleIIL> {
			int a; int b; long c;
			TupleIIL() {  }
			TupleIIL(int a, int b, long c) { this.a = a; this.b = b; this.c = c; }
			@Override public String toString() { return "("+a+", "+b+", "+c+")"; }
			@Override public int hashCode() { return Objects.hash(a, b, c); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null || this.getClass() != obj.getClass()) return false;
				TupleIIL that = (TupleIIL) obj;
				if(this.a != that.a || this.b != that.b || this.c != that.c) return false;
				return true;
			}
			@Override
			public int compareTo(TupleIIL that) {
				int c = Integer.compare(this.a, that.a);
				if(c == 0) c = Integer.compare(this.b, that.b);
				if(c == 0) c = Long.compare(this.c, that.c);
				return c;
			}
		}
		class TupleIID implements Comparable<TupleIID> {
			int a; int b; double c;
			TupleIID() {  }
			TupleIID(int a, int b, double c) { this.a = a; this.b = b; this.c = c; }
			@Override public String toString() { return "("+a+", "+b+", "+c+")"; }
			@Override public int hashCode() { return Objects.hash(a, b, c); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null || this.getClass() != obj.getClass()) return false;
				TupleIID that = (TupleIID) obj;
				if(this.a != that.a || this.b != that.b || this.c != that.c) return false;
				return true;
			}
			@Override
			public int compareTo(TupleIID that) {
				int c = Integer.compare(this.a, that.a);
				if(c == 0) c = Integer.compare(this.b, that.b);
				if(c == 0) c = Double.compare(this.c, that.c);
				return c;
			}
		}
		class TupleILI implements Comparable<TupleILI> {
			int a; long b; int c;
			TupleILI() {  }
			TupleILI(int a, long b, int c) { this.a = a; this.b = b; this.c = c; }
			@Override public String toString() { return "("+a+", "+b+", "+c+")"; }
			@Override public int hashCode() { return Objects.hash(a, b, c); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null || this.getClass() != obj.getClass()) return false;
				TupleILI that = (TupleILI) obj;
				if(this.a != that.a || this.b != that.b || this.c != that.c) return false;
				return true;
			}
			@Override
			public int compareTo(TupleILI that) {
				int c = Integer.compare(this.a, that.a);
				if(c == 0) c = Long.compare(this.b, that.b);
				if(c == 0) c = Integer.compare(this.c, that.c);
				return c;
			}
		}
		class TupleILL implements Comparable<TupleILL> {
			int a; long b; long c;
			TupleILL() {  }
			TupleILL(int a, long b, long c) { this.a = a; this.b = b; this.c = c; }
			@Override public String toString() { return "("+a+", "+b+", "+c+")"; }
			@Override public int hashCode() { return Objects.hash(a, b, c); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null || this.getClass() != obj.getClass()) return false;
				TupleILL that = (TupleILL) obj;
				if(this.a != that.a || this.b != that.b || this.c != that.c) return false;
				return true;
			}
			@Override
			public int compareTo(TupleILL that) {
				int c = Integer.compare(this.a, that.a);
				if(c == 0) c = Long.compare(this.b, that.b);
				if(c == 0) c = Long.compare(this.c, that.c);
				return c;
			}
		}
		class TupleILD implements Comparable<TupleILD> {
			int a; long b; double c;
			TupleILD() {  }
			TupleILD(int a, long b, double c) { this.a = a; this.b = b; this.c = c; }
			@Override public String toString() { return "("+a+", "+b+", "+c+")"; }
			@Override public int hashCode() { return Objects.hash(a, b, c); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null || this.getClass() != obj.getClass()) return false;
				TupleILD that = (TupleILD) obj;
				if(this.a != that.a || this.b != that.b || this.c != that.c) return false;
				return true;
			}
			@Override
			public int compareTo(TupleILD that) {
				int c = Integer.compare(this.a, that.a);
				if(c == 0) c = Long.compare(this.b, that.b);
				if(c == 0) c = Double.compare(this.c, that.c);
				return c;
			}
		}
		class TupleIDI implements Comparable<TupleIDI> {
			int a; double b; int c;
			TupleIDI() {  }
			TupleIDI(int a, double b, int c) { this.a = a; this.b = b; this.c = c; }
			@Override public String toString() { return "("+a+", "+b+", "+c+")"; }
			@Override public int hashCode() { return Objects.hash(a, b, c); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null || this.getClass() != obj.getClass()) return false;
				TupleIDI that = (TupleIDI) obj;
				if(this.a != that.a || this.b != that.b || this.c != that.c) return false;
				return true;
			}
			@Override
			public int compareTo(TupleIDI that) {
				int c = Integer.compare(this.a, that.a);
				if(c == 0) c = Double.compare(this.b, that.b);
				if(c == 0) c = Integer.compare(this.c, that.c);
				return c;
			}
		}
		class TupleIDL implements Comparable<TupleIDL> {
			int a; double b; long c;
			TupleIDL() {  }
			TupleIDL(int a, double b, long c) { this.a = a; this.b = b; this.c = c; }
			@Override public String toString() { return "("+a+", "+b+", "+c+")"; }
			@Override public int hashCode() { return Objects.hash(a, b, c); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null || this.getClass() != obj.getClass()) return false;
				TupleIDL that = (TupleIDL) obj;
				if(this.a != that.a || this.b != that.b || this.c != that.c) return false;
				return true;
			}
			@Override
			public int compareTo(TupleIDL that) {
				int c = Integer.compare(this.a, that.a);
				if(c == 0) c = Double.compare(this.b, that.b);
				if(c == 0) c = Long.compare(this.c, that.c);
				return c;
			}
		}
		class TupleIDD implements Comparable<TupleIDD> {
			int a; double b; double c;
			TupleIDD() {  }
			TupleIDD(int a, double b, double c) { this.a = a; this.b = b; this.c = c; }
			@Override public String toString() { return "("+a+", "+b+", "+c+")"; }
			@Override public int hashCode() { return Objects.hash(a, b, c); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null || this.getClass() != obj.getClass()) return false;
				TupleIDD that = (TupleIDD) obj;
				if(this.a != that.a || this.b != that.b || this.c != that.c) return false;
				return true;
			}
			@Override
			public int compareTo(TupleIDD that) {
				int c = Integer.compare(this.a, that.a);
				if(c == 0) c = Double.compare(this.b, that.b);
				if(c == 0) c = Double.compare(this.c, that.c);
				return c;
			}
		}
		class TupleLII implements Comparable<TupleLII> {
			long a; int b; int c;
			TupleLII() {  }
			TupleLII(long a, int b, int c) { this.a = a; this.b = b; this.c = c; }
			@Override public String toString() { return "("+a+", "+b+", "+c+")"; }
			@Override public int hashCode() { return Objects.hash(a, b, c); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null || this.getClass() != obj.getClass()) return false;
				TupleLII that = (TupleLII) obj;
				if(this.a != that.a || this.b != that.b || this.c != that.c) return false;
				return true;
			}
			@Override
			public int compareTo(TupleLII that) {
				int c = Long.compare(this.a, that.a);
				if(c == 0) c = Integer.compare(this.b, that.b);
				if(c == 0) c = Integer.compare(this.c, that.c);
				return c;
			}
		}
		class TupleLIL implements Comparable<TupleLIL> {
			long a; int b; long c;
			TupleLIL() {  }
			TupleLIL(long a, int b, long c) { this.a = a; this.b = b; this.c = c; }
			@Override public String toString() { return "("+a+", "+b+", "+c+")"; }
			@Override public int hashCode() { return Objects.hash(a, b, c); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null || this.getClass() != obj.getClass()) return false;
				TupleLIL that = (TupleLIL) obj;
				if(this.a != that.a || this.b != that.b || this.c != that.c) return false;
				return true;
			}
			@Override
			public int compareTo(TupleLIL that) {
				int c = Long.compare(this.a, that.a);
				if(c == 0) c = Integer.compare(this.b, that.b);
				if(c == 0) c = Long.compare(this.c, that.c);
				return c;
			}
		}
		class TupleLID implements Comparable<TupleLID> {
			long a; int b; double c;
			TupleLID() {  }
			TupleLID(long a, int b, double c) { this.a = a; this.b = b; this.c = c; }
			@Override public String toString() { return "("+a+", "+b+", "+c+")"; }
			@Override public int hashCode() { return Objects.hash(a, b, c); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null || this.getClass() != obj.getClass()) return false;
				TupleLID that = (TupleLID) obj;
				if(this.a != that.a || this.b != that.b || this.c != that.c) return false;
				return true;
			}
			@Override
			public int compareTo(TupleLID that) {
				int c = Long.compare(this.a, that.a);
				if(c == 0) c = Integer.compare(this.b, that.b);
				if(c == 0) c = Double.compare(this.c, that.c);
				return c;
			}
		}
		class TupleLLI implements Comparable<TupleLLI> {
			long a; long b; int c;
			TupleLLI() {  }
			TupleLLI(long a, long b, int c) { this.a = a; this.b = b; this.c = c; }
			@Override public String toString() { return "("+a+", "+b+", "+c+")"; }
			@Override public int hashCode() { return Objects.hash(a, b, c); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null || this.getClass() != obj.getClass()) return false;
				TupleLLI that = (TupleLLI) obj;
				if(this.a != that.a || this.b != that.b || this.c != that.c) return false;
				return true;
			}
			@Override
			public int compareTo(TupleLLI that) {
				int c = Long.compare(this.a, that.a);
				if(c == 0) c = Long.compare(this.b, that.b);
				if(c == 0) c = Integer.compare(this.c, that.c);
				return c;
			}
		}
		class TupleLLL implements Comparable<TupleLLL> {
			long a; long b; long c;
			TupleLLL() {  }
			TupleLLL(long a, long b, long c) { this.a = a; this.b = b; this.c = c; }
			@Override public String toString() { return "("+a+", "+b+", "+c+")"; }
			@Override public int hashCode() { return Objects.hash(a, b, c); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null || this.getClass() != obj.getClass()) return false;
				TupleLLL that = (TupleLLL) obj;
				if(this.a != that.a || this.b != that.b || this.c != that.c) return false;
				return true;
			}
			@Override
			public int compareTo(TupleLLL that) {
				int c = Long.compare(this.a, that.a);
				if(c == 0) c = Long.compare(this.b, that.b);
				if(c == 0) c = Long.compare(this.c, that.c);
				return c;
			}
		}
		class TupleLLD implements Comparable<TupleLLD> {
			long a; long b; double c;
			TupleLLD() {  }
			TupleLLD(long a, long b, double c) { this.a = a; this.b = b; this.c = c; }
			@Override public String toString() { return "("+a+", "+b+", "+c+")"; }
			@Override public int hashCode() { return Objects.hash(a, b, c); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null || this.getClass() != obj.getClass()) return false;
				TupleLLD that = (TupleLLD) obj;
				if(this.a != that.a || this.b != that.b || this.c != that.c) return false;
				return true;
			}
			@Override
			public int compareTo(TupleLLD that) {
				int c = Long.compare(this.a, that.a);
				if(c == 0) c = Long.compare(this.b, that.b);
				if(c == 0) c = Double.compare(this.c, that.c);
				return c;
			}
		}
		class TupleLDI implements Comparable<TupleLDI> {
			long a; double b; int c;
			TupleLDI() {  }
			TupleLDI(long a, double b, int c) { this.a = a; this.b = b; this.c = c; }
			@Override public String toString() { return "("+a+", "+b+", "+c+")"; }
			@Override public int hashCode() { return Objects.hash(a, b, c); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null || this.getClass() != obj.getClass()) return false;
				TupleLDI that = (TupleLDI) obj;
				if(this.a != that.a || this.b != that.b || this.c != that.c) return false;
				return true;
			}
			@Override
			public int compareTo(TupleLDI that) {
				int c = Long.compare(this.a, that.a);
				if(c == 0) c = Double.compare(this.b, that.b);
				if(c == 0) c = Integer.compare(this.c, that.c);
				return c;
			}
		}
		class TupleLDL implements Comparable<TupleLDL> {
			long a; double b; long c;
			TupleLDL() {  }
			TupleLDL(long a, double b, long c) { this.a = a; this.b = b; this.c = c; }
			@Override public String toString() { return "("+a+", "+b+", "+c+")"; }
			@Override public int hashCode() { return Objects.hash(a, b, c); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null || this.getClass() != obj.getClass()) return false;
				TupleLDL that = (TupleLDL) obj;
				if(this.a != that.a || this.b != that.b || this.c != that.c) return false;
				return true;
			}
			@Override
			public int compareTo(TupleLDL that) {
				int c = Long.compare(this.a, that.a);
				if(c == 0) c = Double.compare(this.b, that.b);
				if(c == 0) c = Long.compare(this.c, that.c);
				return c;
			}
		}
		class TupleLDD implements Comparable<TupleLDD> {
			long a; double b; double c;
			TupleLDD() {  }
			TupleLDD(long a, double b, double c) { this.a = a; this.b = b; this.c = c; }
			@Override public String toString() { return "("+a+", "+b+", "+c+")"; }
			@Override public int hashCode() { return Objects.hash(a, b, c); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null || this.getClass() != obj.getClass()) return false;
				TupleLDD that = (TupleLDD) obj;
				if(this.a != that.a || this.b != that.b || this.c != that.c) return false;
				return true;
			}
			@Override
			public int compareTo(TupleLDD that) {
				int c = Long.compare(this.a, that.a);
				if(c == 0) c = Double.compare(this.b, that.b);
				if(c == 0) c = Double.compare(this.c, that.c);
				return c;
			}
		}
		class TupleDII implements Comparable<TupleDII> {
			double a; int b; int c;
			TupleDII() {  }
			TupleDII(double a, int b, int c) { this.a = a; this.b = b; this.c = c; }
			@Override public String toString() { return "("+a+", "+b+", "+c+")"; }
			@Override public int hashCode() { return Objects.hash(a, b, c); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null || this.getClass() != obj.getClass()) return false;
				TupleDII that = (TupleDII) obj;
				if(this.a != that.a || this.b != that.b || this.c != that.c) return false;
				return true;
			}
			@Override
			public int compareTo(TupleDII that) {
				int c = Double.compare(this.a, that.a);
				if(c == 0) c = Integer.compare(this.b, that.b);
				if(c == 0) c = Integer.compare(this.c, that.c);
				return c;
			}
		}
		class TupleDIL implements Comparable<TupleDIL> {
			double a; int b; long c;
			TupleDIL() {  }
			TupleDIL(double a, int b, long c) { this.a = a; this.b = b; this.c = c; }
			@Override public String toString() { return "("+a+", "+b+", "+c+")"; }
			@Override public int hashCode() { return Objects.hash(a, b, c); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null || this.getClass() != obj.getClass()) return false;
				TupleDIL that = (TupleDIL) obj;
				if(this.a != that.a || this.b != that.b || this.c != that.c) return false;
				return true;
			}
			@Override
			public int compareTo(TupleDIL that) {
				int c = Double.compare(this.a, that.a);
				if(c == 0) c = Integer.compare(this.b, that.b);
				if(c == 0) c = Long.compare(this.c, that.c);
				return c;
			}
		}
		class TupleDID implements Comparable<TupleDID> {
			double a; int b; double c;
			TupleDID() {  }
			TupleDID(double a, int b, double c) { this.a = a; this.b = b; this.c = c; }
			@Override public String toString() { return "("+a+", "+b+", "+c+")"; }
			@Override public int hashCode() { return Objects.hash(a, b, c); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null || this.getClass() != obj.getClass()) return false;
				TupleDID that = (TupleDID) obj;
				if(this.a != that.a || this.b != that.b || this.c != that.c) return false;
				return true;
			}
			@Override
			public int compareTo(TupleDID that) {
				int c = Double.compare(this.a, that.a);
				if(c == 0) c = Integer.compare(this.b, that.b);
				if(c == 0) c = Double.compare(this.c, that.c);
				return c;
			}
		}
		class TupleDLI implements Comparable<TupleDLI> {
			double a; long b; int c;
			TupleDLI() {  }
			TupleDLI(double a, long b, int c) { this.a = a; this.b = b; this.c = c; }
			@Override public String toString() { return "("+a+", "+b+", "+c+")"; }
			@Override public int hashCode() { return Objects.hash(a, b, c); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null || this.getClass() != obj.getClass()) return false;
				TupleDLI that = (TupleDLI) obj;
				if(this.a != that.a || this.b != that.b || this.c != that.c) return false;
				return true;
			}
			@Override
			public int compareTo(TupleDLI that) {
				int c = Double.compare(this.a, that.a);
				if(c == 0) c = Long.compare(this.b, that.b);
				if(c == 0) c = Integer.compare(this.c, that.c);
				return c;
			}
		}
		class TupleDLL implements Comparable<TupleDLL> {
			double a; long b; long c;
			TupleDLL() {  }
			TupleDLL(double a, long b, long c) { this.a = a; this.b = b; this.c = c; }
			@Override public String toString() { return "("+a+", "+b+", "+c+")"; }
			@Override public int hashCode() { return Objects.hash(a, b, c); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null || this.getClass() != obj.getClass()) return false;
				TupleDLL that = (TupleDLL) obj;
				if(this.a != that.a || this.b != that.b || this.c != that.c) return false;
				return true;
			}
			@Override
			public int compareTo(TupleDLL that) {
				int c = Double.compare(this.a, that.a);
				if(c == 0) c = Long.compare(this.b, that.b);
				if(c == 0) c = Long.compare(this.c, that.c);
				return c;
			}
		}
		class TupleDLD implements Comparable<TupleDLD> {
			double a; long b; double c;
			TupleDLD() {  }
			TupleDLD(double a, long b, double c) { this.a = a; this.b = b; this.c = c; }
			@Override public String toString() { return "("+a+", "+b+", "+c+")"; }
			@Override public int hashCode() { return Objects.hash(a, b, c); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null || this.getClass() != obj.getClass()) return false;
				TupleDLD that = (TupleDLD) obj;
				if(this.a != that.a || this.b != that.b || this.c != that.c) return false;
				return true;
			}
			@Override
			public int compareTo(TupleDLD that) {
				int c = Double.compare(this.a, that.a);
				if(c == 0) c = Long.compare(this.b, that.b);
				if(c == 0) c = Double.compare(this.c, that.c);
				return c;
			}
		}
		class TupleDDI implements Comparable<TupleDDI> {
			double a; double b; int c;
			TupleDDI() {  }
			TupleDDI(double a, double b, int c) { this.a = a; this.b = b; this.c = c; }
			@Override public String toString() { return "("+a+", "+b+", "+c+")"; }
			@Override public int hashCode() { return Objects.hash(a, b, c); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null || this.getClass() != obj.getClass()) return false;
				TupleDDI that = (TupleDDI) obj;
				if(this.a != that.a || this.b != that.b || this.c != that.c) return false;
				return true;
			}
			@Override
			public int compareTo(TupleDDI that) {
				int c = Double.compare(this.a, that.a);
				if(c == 0) c = Double.compare(this.b, that.b);
				if(c == 0) c = Integer.compare(this.c, that.c);
				return c;
			}
		}
		class TupleDDL implements Comparable<TupleDDL> {
			double a; double b; long c;
			TupleDDL() {  }
			TupleDDL(double a, double b, long c) { this.a = a; this.b = b; this.c = c; }
			@Override public String toString() { return "("+a+", "+b+", "+c+")"; }
			@Override public int hashCode() { return Objects.hash(a, b, c); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null || this.getClass() != obj.getClass()) return false;
				TupleDDL that = (TupleDDL) obj;
				if(this.a != that.a || this.b != that.b || this.c != that.c) return false;
				return true;
			}
			@Override
			public int compareTo(TupleDDL that) {
				int c = Double.compare(this.a, that.a);
				if(c == 0) c = Double.compare(this.b, that.b);
				if(c == 0) c = Long.compare(this.c, that.c);
				return c;
			}
		}
		class TupleDDD implements Comparable<TupleDDD> {
			double a; double b; double c;
			TupleDDD() {  }
			TupleDDD(double a, double b, double c) { this.a = a; this.b = b; this.c = c; }
			@Override public String toString() { return "("+a+", "+b+", "+c+")"; }
			@Override public int hashCode() { return Objects.hash(a, b, c); }
			@Override
			public boolean equals(Object obj) {
				if(this == obj) return true;
				if(obj == null || this.getClass() != obj.getClass()) return false;
				TupleDDD that = (TupleDDD) obj;
				if(this.a != that.a || this.b != that.b || this.c != that.c) return false;
				return true;
			}
			@Override
			public int compareTo(TupleDDD that) {
				int c = Double.compare(this.a, that.a);
				if(c == 0) c = Double.compare(this.b, that.b);
				if(c == 0) c = Double.compare(this.c, that.c);
				return c;
			}
		}

public void solve() {
	int h = ni();
	int w = ni();
	Grids g = new Grids(h, w);
	boolean b[][] = nb(h, w, '#');
	g.init(b);
	MaxFlow mf = new MaxFlow(h * w + 2);
	for(int i = 0; i < h; i ++) {
		for(int j = 0; j < w - 1; j ++) {
			if(!g.get(i, j).b && !g.get(i, j + 1).b) {
				if((i + j) % 2 == 0) mf.addEdge(new Edge(g.get(i, j).i, g.get(i, j + 1).i, 1));
				else mf.addEdge(new Edge(g.get(i, j + 1).i, g.get(i, j).i, 1));
			}
		}
	}
	for(int i = 0; i < h - 1; i ++) {
		for(int j = 0; j < w; j ++) {
			if(!g.get(i, j).b && !g.get(i + 1, j).b) {
				if((i + j) % 2 == 0) mf.addEdge(new Edge(g.get(i, j).i, g.get(i + 1, j).i, 1));
				else mf.addEdge(new Edge(g.get(i + 1, j).i, g.get(i, j).i, 1));
			}
		}
	}
	for(int i = 0; i < h; i ++) {
		for(int j = 0; j < w; j ++) {
			if(!g.get(i, j).b) {
				if((i + j) % 2 == 0) mf.addEdge(new Edge(h * w, g.get(i, j).i, 1));
				else mf.addEdge(new Edge(g.get(i, j).i, h * w + 1, 1));
			}
		}
	}
	prtln(mf.maxFlow(h * w, h * w + 1));
	char ans[][] = new char[h][w];
	for(int i = 0; i < h; i ++) for(int j = 0; j < w; j ++) ans[i][j] = g.get(i, j).b ? '#' : '.';
	for(MaxFlow.CapEdge e : mf.edgesList) {
		if(e.getFlow() == 0) continue;
		if(e.source >= h * w) continue;
		if(e.target >= h * w) continue;
		Grid source = g.get(e.source);
		Grid target = g.get(e.target);
		if(source.x == target.x) {
			if(source.y < target.y) { ans[source.x][source.y] = '>'; ans[target.x][target.y] = '<'; }
			if(source.y > target.y) { ans[source.x][source.y] = '<'; ans[target.x][target.y] = '>'; }
		}
		if(source.y == target.y) {
			if(source.x < target.x) { ans[source.x][source.y] = 'v'; ans[target.x][target.y] = '^'; }
			if(source.x > target.x) { ans[source.x][source.y] = '^'; ans[target.x][target.y] = 'v'; }
		}
	}
	prtln(ans);
}

class MaxFlow { // V=numNode, E=numEdge
	class CapEdge {
		int source;
		int target;
		long cap;
		int rev;
		CapEdge(int source, int target, long cap, int rev) {
			this.source = source;
			this.target = target;
			this.cap = cap;
			this.rev = rev;
		}
		long getFlow() { return edges[target][rev].cap; }

		@Override
		public String toString() { return source+" - "+cap+" -> "+target; }
		@Override
		public int hashCode() { return Objects.hash(source, target); }
		@Override
		public boolean equals(Object obj) {
			if(this == obj) return true;
			if(obj == null) return false;
			if(this.getClass() != obj.getClass()) return false;
			Edge that = (Edge) obj;
			if(this.source != that.source) return false;
			if(this.target != that.target) return false;
			return true;
		}
	}

	int numNode;
	int numEdge;
	ArrayList<CapEdge> edgesList;
	int len[];
	CapEdge edges[][];

	MaxFlow(int numNode) { // O(V^2)
		this.numNode = numNode;

		numEdge = 0;
		edgesList = new ArrayList<>();
		len = new int[numNode];
		edges = new CapEdge[numNode][];
	}

	int addEdge(Edge e) { return addEdge(e.source, e.target, e.cost); } // O(1)
	int addEdge(int source, int target, long cap) { // O(1)
		CapEdge e = new CapEdge(source, target, cap, len[target]);
		len[source] ++;
		len[target] ++;
		edgesList.add(e);
		return numEdge ++;
	}

	void changeEdge(int i, long newCap, long newFlow) {
		if(newFlow > newCap) return;
		CapEdge e = edgesList.get(i);
		CapEdge er = edges[e.target][e.rev];
		e.cap = newCap - newFlow;
		er.cap = newFlow;
	}

	void init() {
		for(int i = 0; i < numNode; i++) edges[i] = new CapEdge[len[i]];
		int idx[] = new int[numNode];
		for(CapEdge e : edgesList) {
			edges[e.target][idx[e.target] ++] = new CapEdge(e.target, e.source, 0, idx[e.source]);
			edges[e.source][idx[e.source] ++] = e;
		}
	}


	long maxFlow(int s, int t) { return flow(s, t, INF); } // O(V^2E) // O(E竏啖) with bipartite graph
	long flow(int s, int t, long flowLimit) {
		int iter[] = new int[numNode];
		init();
		long flow = 0;
		while(true) {
			long depth[] = dinicBFS(s, t);
			if(isINF(depth[t])) return flow;
			fill(iter, 0);
			long f = 0;
			do {
				f = dinicDFS(s, t, flowLimit - flow, depth, iter);
				flow += f;
			}while(f > 0);
		}
	}

	long[] dinicBFS(int s, int t) { // O(V)
		long depth[] = new long[numNode];
		fill(depth, INF);

		int dq[] = new int[numNode];
		int size = 0;
		int ptr = 0;

		dq[size ++] = s;
		depth[s] = 0;
		while(ptr < size) {
			int crt = dq[ptr ++];
			for(CapEdge e : edges[crt]) {
				if(e.cap <= 0 || !isINF(depth[e.target])) continue;
				depth[e.target] = depth[e.source] + 1;
				if(e.target == t) return depth;
				dq[size ++] = e.target;
			}
		}
		return depth;
	}

	long dinicDFS(int v, int t, long f, long[] depth, int[] iter) { // O(VE)
		if(v == t) return f;
		for(; iter[v] < len[v]; iter[v] ++) {
			CapEdge e = edges[v][iter[v]];
			if(e.cap <= 0 || depth[e.source] >= depth[e.target]) continue;
			long d = dinicDFS(e.target, t, min(f, e.cap), depth, iter);
			if(d <= 0) continue;
			e.cap -= d;
			edges[e.target][e.rev].cap += d;
			return d;
		}
		return 0;
	}


	long fordFulkersonMaxFlow(int s, int t) { return fordFulkersonFlow(s, t, INF); } // O(Ef)
	long fordFulkersonFlow(int s, int t, long flowLimit) {
		init();
		boolean[] used = new boolean[numNode];
		long flow = 0;
		while(true) {
			fill(used, false);
			long f = fordFulkersonDFS(s, t, flowLimit - flow, used);
			if(f <= 0) return flow;
			flow += f;
		}
	}

	long fordFulkersonDFS(int v, int t, long f, boolean[] used) {
		if(v == t) return f;
		used[v] = true;
		for(CapEdge e : edges[v]) {
			if(e.cap <= 0 || used[e.target]) continue;
			long d = fordFulkersonDFS(e.target, t, min(f, e.cap), used);
			if(d <= 0) continue;
			e.cap -= d;
			edges[e.target][e.rev].cap += d;
			return d;
		}
		return 0;
	}


	boolean[] minCut(int s) {
		boolean[] reachable = new boolean[numNode];
		int[] dq = new int[numNode];
		int ptr = 0;
		dq[ptr ++] = s;
		reachable[s] = true;
		while(ptr > 0) {
			int crt = dq[-- ptr];
			for(CapEdge e : edges[crt]) {
				if(reachable[e.target] || e.cap <= 0) continue;
				reachable[e.target] = true;
				dq[ptr ++] = e.target;
			}
		}
		return reachable;
	}

}

	}
}