Problem: It's holiday. Mashmokh and his boss, Bimokh, are playing a game invented by Mashmokh. 

In this game Mashmokh writes sequence of n distinct integers on the board. Then Bimokh makes several (possibly zero) moves. On the first move he removes the first and the second integer from from the board, on the second move he removes the first and the second integer of the remaining sequence from the board, and so on. Bimokh stops when the board contains less than two numbers. When Bimokh removes numbers x and y from the board, he gets gcd(x, y) points. At the beginning of the game Bimokh has zero points.

Mashmokh wants to win in the game. For this reason he wants his boss to get exactly k points in total. But the guy doesn't know how choose the initial sequence in the right way. 

Please, help him. Find n distinct integers a1, a2, ..., an such that his boss will score exactly k points. Also Mashmokh can't memorize too huge numbers. Therefore each of these integers must be at most 109.

Input

The first line of input contains two space-separated integers n, k (1 ≤ n ≤ 105; 0 ≤ k ≤ 108).

Output

If such sequence doesn't exist output -1 otherwise output n distinct space-separated integers a1, a2, ..., an (1 ≤ ai ≤ 109).

Examples

Input

5 2


Output

1 2 3 4 5


Input

5 3

Output

2 4 3 7 1

Input

7 2


Output

-1

Note

gcd(x, y) is greatest common divisor of x and y.

Solution (in JAVA):

Solution:
import java.util.*;
import java.io.*;
import java.math.*;
import java.lang.reflect.*;

public class Main {
	public static void main(String[] args) {
		InputStream inputStream = System.in;
		OutputStream outputStream = System.out;
		InputReader in = new InputReader(inputStream);
		OutputWriter out = new OutputWriter(outputStream);
		Task solver = new TaskC();
		solver.solve(1, in, out);
		out.close();
	}
}

interface Task {
	public void solve(int testNumber, InputReader in, OutputWriter out);
}

class TaskA implements Task {

	public void solve(int testNumber, InputReader in, OutputWriter out) {
		out.printLine(in.readInt(), 0, 0);
	}

}

class TaskB implements Task {

	long[] array;
	boolean[] isPrime;
	TreeSet<Long> set;
	
	public void solve(int testNumber, InputReader in, OutputWriter out) {
		int n=in.readInt();
		array=IOUtils.readLongArray(in, n);
		isPrime=IntegerUtils.generatePrimalityTable(1000001);
		set=new TreeSet<Long>();
		for (int i=1; i<=1000000; i++) if (isPrime[i]) set.add(1L*i*i);
		for (long i: array) out.printLine(set.contains(i)?"YES":"NO");
	}

}

class TaskC implements Task {

	int[] primes;
	
	public void solve(int testNumber, InputReader in, OutputWriter out) {
		primes=IntegerUtils.generatePrimes(10000000);
		int n=in.readInt(), k=in.readInt();
		if (k<n/2 || n<2 && k>0) {
			out.printLine(-1);
			return;
		}
		if (k==0) {
			out.printLine(1);
			return;
		}
		k-=(n-2)/2;
		if (k==1) n+=2;
		else out.print(2*k, 3*k, "");
		for (int i=0; i<n-2; i++) out.print(primes[i], "");
	}

}

class TaskD implements Task {

	public void solve(int testNumber, InputReader in, OutputWriter out) {
		int n=in.readInt();
		out.printLine(1L*(n/2+1)*((n+1)/2+1)%1000000009);
	}
}

class TaskE implements Task {

	public void solve(int testNumber, InputReader in, OutputWriter out) {

	}

}

class TaskF implements Task {

	public void solve(int testNumber, InputReader in, OutputWriter out) {

	}

}

class OutputWriter {
	private final PrintWriter writer;

	public OutputWriter(OutputStream outputStream) {
		writer = new PrintWriter(new BufferedWriter(new OutputStreamWriter(
				outputStream)));
	}

	public OutputWriter(Writer writer) {
		this.writer = new PrintWriter(writer);
	}

	public void print(char[] array) {
		writer.print(array);
	}

	public void print(Object... objects) {
		for (int i = 0; i < objects.length; i++) {
			if (i != 0)
				writer.print(' ');
			writer.print(objects[i]);
		}
	}

	public void print(int[] array) {
		for (int i = 0; i < array.length; i++) {
			if (i != 0)
				writer.print(' ');
			writer.print(array[i]);
		}
	}

	public void print(long[] array) {
		for (int i = 0; i < array.length; i++) {
			if (i != 0)
				writer.print(' ');
			writer.print(array[i]);
		}
	}

	public void print(Collection<Integer> collection) {
		boolean first = true;
		for (int value : collection) {
			if (first)
				first = false;
			else
				writer.print(' ');
			writer.print(value);
		}
	}

	public void printLine(int[] array) {
		print(array);
		writer.println();
	}

	public void printLine(long[] array) {
		print(array);
		writer.println();
	}

	public void printLine(Collection<Integer> collection) {
		print(collection);
		writer.println();
	}

	public void printLine() {
		writer.println();
	}

	public void printLine(Object... objects) {
		print(objects);
		writer.println();
	}

	public void print(char i) {
		writer.print(i);
	}

	public void printLine(char i) {
		writer.println(i);
	}

	public void printLine(char[] array) {
		writer.println(array);
	}

	public void printFormat(String format, Object... objects) {
		writer.printf(format, objects);
	}

	public void close() {
		writer.close();
	}

	public void flush() {
		writer.flush();
	}

	public void print(long i) {
		writer.print(i);
	}

	public void printLine(long i) {
		writer.println(i);
	}

	public void print(int i) {
		writer.print(i);
	}

	public void printLine(int i) {
		writer.println(i);
	}
}

class InputReader {
	private boolean finished = false;

	private InputStream stream;
	private byte[] buf = new byte[1024];
	private int curChar;
	private int numChars;
	private SpaceCharFilter filter;

	public InputReader(InputStream stream) {
		this.stream = stream;
	}

	public int read() {
		if (numChars == -1)
			throw new InputMismatchException();
		if (curChar >= numChars) {
			curChar = 0;
			try {
				numChars = stream.read(buf);
			} catch (IOException e) {
				throw new InputMismatchException();
			}
			if (numChars <= 0)
				return -1;
		}
		return buf[curChar++];
	}

	public int peek() {
		if (numChars == -1)
			return -1;
		if (curChar >= numChars) {
			curChar = 0;
			try {
				numChars = stream.read(buf);
			} catch (IOException e) {
				return -1;
			}
			if (numChars <= 0)
				return -1;
		}
		return buf[curChar];
	}

	public int readInt() {
		int c = read();
		while (isSpaceChar(c))
			c = read();
		int sgn = 1;
		if (c == '-') {
			sgn = -1;
			c = read();
		}
		int res = 0;
		do {
			if (c < '0' || c > '9')
				throw new InputMismatchException();
			res *= 10;
			res += c - '0';
			c = read();
		} while (!isSpaceChar(c));
		return res * sgn;
	}

	public long readLong() {
		int c = read();
		while (isSpaceChar(c))
			c = read();
		int sgn = 1;
		if (c == '-') {
			sgn = -1;
			c = read();
		}
		long res = 0;
		do {
			if (c < '0' || c > '9')
				throw new InputMismatchException();
			res *= 10;
			res += c - '0';
			c = read();
		} while (!isSpaceChar(c));
		return res * sgn;
	}

	public String readString() {
		int c = read();
		while (isSpaceChar(c))
			c = read();
		StringBuilder res = new StringBuilder();
		do {
			if (Character.isValidCodePoint(c))
				res.appendCodePoint(c);
			c = read();
		} while (!isSpaceChar(c));
		return res.toString();
	}

	public boolean isSpaceChar(int c) {
		if (filter != null)
			return filter.isSpaceChar(c);
		return isWhitespace(c);
	}

	public static boolean isWhitespace(int c) {
		return c == ' ' || c == '\n' || c == '\r' || c == '\t' || c == -1;
	}

	private String readLine0() {
		StringBuilder buf = new StringBuilder();
		int c = read();
		while (c != '\n' && c != -1) {
			if (c != '\r')
				buf.appendCodePoint(c);
			c = read();
		}
		return buf.toString();
	}

	public String readLine() {
		String s = readLine0();
		while (s.trim().length() == 0)
			s = readLine0();
		return s;
	}

	public String readLine(boolean ignoreEmptyLines) {
		if (ignoreEmptyLines)
			return readLine();
		else
			return readLine0();
	}

	public BigInteger readBigInteger() {
		try {
			return new BigInteger(readString());
		} catch (NumberFormatException e) {
			throw new InputMismatchException();
		}
	}

	public char readCharacter() {
		int c = read();
		while (isSpaceChar(c))
			c = read();
		return (char) c;
	}

	public double readDouble() {
		int c = read();
		while (isSpaceChar(c))
			c = read();
		int sgn = 1;
		if (c == '-') {
			sgn = -1;
			c = read();
		}
		double res = 0;
		while (!isSpaceChar(c) && c != '.') {
			if (c == 'e' || c == 'E')
				return res * Math.pow(10, readInt());
			if (c < '0' || c > '9')
				throw new InputMismatchException();
			res *= 10;
			res += c - '0';
			c = read();
		}
		if (c == '.') {
			c = read();
			double m = 1;
			while (!isSpaceChar(c)) {
				if (c == 'e' || c == 'E')
					return res * Math.pow(10, readInt());
				if (c < '0' || c > '9')
					throw new InputMismatchException();
				m /= 10;
				res += (c - '0') * m;
				c = read();
			}
		}
		return res * sgn;
	}

	public boolean isExhausted() {
		int value;
		while (isSpaceChar(value = peek()) && value != -1)
			read();
		return value == -1;
	}

	public String next() {
		return readString();
	}

	public SpaceCharFilter getFilter() {
		return filter;
	}

	public void setFilter(SpaceCharFilter filter) {
		this.filter = filter;
	}

	public interface SpaceCharFilter {
		public boolean isSpaceChar(int ch);
	}
}

class IOUtils {
	public static Pair<Integer, Integer> readIntPair(InputReader in) {
		int first = in.readInt();
		int second = in.readInt();
		return Pair.makePair(first, second);
	}

	public static Pair<Long, Long> readLongPair(InputReader in) {
		long first = in.readLong();
		long second = in.readLong();
		return Pair.makePair(first, second);
	}

	public static int[] readIntArray(InputReader in, int size) {
		int[] array = new int[size];
		for (int i = 0; i < size; i++)
			array[i] = in.readInt();
		return array;
	}

	public static long[] readLongArray(InputReader in, int size) {
		long[] array = new long[size];
		for (int i = 0; i < size; i++)
			array[i] = in.readLong();
		return array;
	}

	public static double[] readDoubleArray(InputReader in, int size) {
		double[] array = new double[size];
		for (int i = 0; i < size; i++)
			array[i] = in.readDouble();
		return array;
	}

	public static String[] readStringArray(InputReader in, int size) {
		String[] array = new String[size];
		for (int i = 0; i < size; i++)
			array[i] = in.readString();
		return array;
	}

	public static char[] readCharArray(InputReader in, int size) {
		char[] array = new char[size];
		for (int i = 0; i < size; i++)
			array[i] = in.readCharacter();
		return array;
	}

	public static Pair<Integer, Integer>[] readIntPairArray(InputReader in,
			int size) {
		@SuppressWarnings({ "unchecked" })
		Pair<Integer, Integer>[] result = new Pair[size];
		for (int i = 0; i < size; i++)
			result[i] = readIntPair(in);
		return result;
	}

	public static Pair<Long, Long>[] readLongPairArray(InputReader in, int size) {
		@SuppressWarnings({ "unchecked" })
		Pair<Long, Long>[] result = new Pair[size];
		for (int i = 0; i < size; i++)
			result[i] = readLongPair(in);
		return result;
	}

	public static void readIntArrays(InputReader in, int[]... arrays) {
		for (int i = 0; i < arrays[0].length; i++) {
			for (int j = 0; j < arrays.length; j++)
				arrays[j][i] = in.readInt();
		}
	}

	public static void readLongArrays(InputReader in, long[]... arrays) {
		for (int i = 0; i < arrays[0].length; i++) {
			for (int j = 0; j < arrays.length; j++)
				arrays[j][i] = in.readLong();
		}
	}

	public static void readDoubleArrays(InputReader in, double[]... arrays) {
		for (int i = 0; i < arrays[0].length; i++) {
			for (int j = 0; j < arrays.length; j++)
				arrays[j][i] = in.readDouble();
		}
	}

	public static char[][] readTable(InputReader in, int rowCount,
			int columnCount) {
		char[][] table = new char[rowCount][];
		for (int i = 0; i < rowCount; i++)
			table[i] = readCharArray(in, columnCount);
		return table;
	}

	public static int[][] readIntTable(InputReader in, int rowCount,
			int columnCount) {
		int[][] table = new int[rowCount][];
		for (int i = 0; i < rowCount; i++)
			table[i] = readIntArray(in, columnCount);
		return table;
	}

	public static double[][] readDoubleTable(InputReader in, int rowCount,
			int columnCount) {
		double[][] table = new double[rowCount][];
		for (int i = 0; i < rowCount; i++)
			table[i] = readDoubleArray(in, columnCount);
		return table;
	}

	public static long[][] readLongTable(InputReader in, int rowCount,
			int columnCount) {
		long[][] table = new long[rowCount][];
		for (int i = 0; i < rowCount; i++)
			table[i] = readLongArray(in, columnCount);
		return table;
	}

	public static String[][] readStringTable(InputReader in, int rowCount,
			int columnCount) {
		String[][] table = new String[rowCount][];
		for (int i = 0; i < rowCount; i++)
			table[i] = readStringArray(in, columnCount);
		return table;
	}

	public static String readText(InputReader in) {
		StringBuilder result = new StringBuilder();
		while (true) {
			int character = in.read();
			if (character == '\r')
				continue;
			if (character == -1)
				break;
			result.append((char) character);
		}
		return result.toString();
	}

	public static void readStringArrays(InputReader in, String[]... arrays) {
		for (int i = 0; i < arrays[0].length; i++) {
			for (int j = 0; j < arrays.length; j++)
				arrays[j][i] = in.readString();
		}
	}

	public static void printTable(OutputWriter out, char[][] table) {
		for (char[] row : table)
			out.printLine(new String(row));
	}
}

class ArrayUtils {
	private static int[] tempInt = new int[0];
	private static long[] tempLong = new long[0];

	public static Integer[] generateOrder(int size) {
		Integer[] order = new Integer[size];
		for (int i = 0; i < size; i++)
			order[i] = i;
		return order;
	}

	public static void fill(short[][] array, short value) {
		for (short[] row : array)
			Arrays.fill(row, value);
	}

	public static void fill(long[][] array, long value) {
		for (long[] row : array)
			Arrays.fill(row, value);
	}

	public static void fill(double[][] array, double value) {
		for (double[] row : array)
			Arrays.fill(row, value);
	}

	public static void fill(double[][][] array, double value) {
		for (double[][] row : array)
			fill(row, value);
	}

	public static void fill(double[][][][] array, double value) {
		for (double[][][] row : array)
			fill(row, value);
	}

	public static void fill(double[][][][][] array, double value) {
		for (double[][][][] row : array)
			fill(row, value);
	}

	public static void fill(long[][][] array, long value) {
		for (long[][] row : array)
			fill(row, value);
	}

	public static void fill(long[][][][] array, long value) {
		for (long[][][] row : array)
			fill(row, value);
	}

	public static void fill(long[][][][][] array, long value) {
		for (long[][][][] row : array)
			fill(row, value);
	}

	public static void fillColumn(long[][] array, int index, long value) {
		for (long[] row : array)
			row[index] = value;
	}

	public static void fillColumn(int[][] array, int index, int value) {
		for (int[] row : array)
			row[index] = value;
	}

	public static void fill(int[][] array, int value) {
		for (int[] row : array)
			Arrays.fill(row, value);
	}

	public static void fill(boolean[][] array, boolean value) {
		for (boolean[] row : array)
			Arrays.fill(row, value);
	}

	public static void fill(boolean[][][] array, boolean value) {
		for (boolean[][] row : array)
			fill(row, value);
	}

	public static long sumArray(int[] array) {
		long result = 0;
		for (int element : array)
			result += element;
		return result;
	}

	public static int[] range(int from, int to) {
		int[] result = new int[Math.max(from, to) - Math.min(from, to) + 1];
		int index = 0;
		if (to > from) {
			for (int i = from; i <= to; i++)
				result[index++] = i;
		} else {
			for (int i = from; i >= to; i--)
				result[index++] = i;
		}
		return result;
	}

	public static void fill(int[][][] array, int value) {
		for (int[][] subArray : array)
			fill(subArray, value);
	}

	public static void fill(short[][][] array, short value) {
		for (short[][] subArray : array)
			fill(subArray, value);
	}

	public static void fill(int[][][][] array, int value) {
		for (int[][][] subArray : array)
			fill(subArray, value);
	}

	public static void fill(short[][][][] array, short value) {
		for (short[][][] subArray : array)
			fill(subArray, value);
	}

	public static void fill(int[][][][][] array, int value) {
		for (int[][][][] subArray : array)
			fill(subArray, value);
	}

	public static void fill(short[][][][][] array, short value) {
		for (short[][][][] subArray : array)
			fill(subArray, value);
	}

	public static void fill(int[][][][][][] array, int value) {
		for (int[][][][][] subArray : array)
			fill(subArray, value);
	}

	public static void fill(short[][][][][][] array, short value) {
		for (short[][][][][] subArray : array)
			fill(subArray, value);
	}

	public static void fill(int[][][][][][][] array, int value) {
		for (int[][][][][][] subArray : array)
			fill(subArray, value);
	}

	public static void fill(short[][][][][][][] array, short value) {
		for (short[][][][][][] subArray : array)
			fill(subArray, value);
	}

	public static Integer[] order(int size, Comparator<Integer> comparator) {
		Integer[] order = generateOrder(size);
		Arrays.sort(order, comparator);
		return order;
	}

	public static <T> void fill(T[][] array, T value) {
		for (T[] row : array)
			Arrays.fill(row, value);
	}

	public static void fill(char[][] array, char value) {
		for (char[] row : array)
			Arrays.fill(row, value);
	}

	public static void fill(byte[][] array, byte value) {
		for (byte[] row : array)
			Arrays.fill(row, value);
	}

	public static void fill(byte[][][] array, byte value) {
		for (byte[][] row : array)
			fill(row, value);
	}

	public static void fill(byte[][][][] array, byte value) {
		for (byte[][][] row : array)
			fill(row, value);
	}

	public static long multiply(int[] first, int[] second) {
		long result = 0;
		for (int i = 0; i < first.length; i++)
			result += (long) first[i] * second[i];
		return result;
	}

	public static int[] createOrder(int size) {
		int[] order = new int[size];
		for (int i = 0; i < size; i++)
			order[i] = i;
		return order;
	}

	public static int[] sort(int[] array, IntComparator comparator) {
		return sort(array, 0, array.length, comparator);
	}

	public static int[] sort(int[] array, int from, int to,
			IntComparator comparator) {
		Integer[] intArray = new Integer[to - from];
		for (int i = from; i < to; i++)
			intArray[i - from] = array[i];
		Arrays.sort(intArray, comparator);
		for (int i=from; i<to; i++) array[i]=intArray[i-from];
		return array;
	}

	private static void ensureCapacityInt(int size) {
		if (tempInt.length >= size)
			return;
		size = Math.max(size, tempInt.length << 1);
		tempInt = new int[size];
	}

	private static void ensureCapacityLong(int size) {
		if (tempLong.length >= size)
			return;
		size = Math.max(size, tempLong.length << 1);
		tempLong = new long[size];
	}

	private static void sortImpl(int[] array, int from, int to, int[] temp,
			int fromTemp, int toTemp, IntComparator comparator) {
		if (to - from <= 1)
			return;
		int middle = (to - from) >> 1;
		int tempMiddle = fromTemp + middle;
		sortImpl(temp, fromTemp, tempMiddle, array, from, from + middle,
				comparator);
		sortImpl(temp, tempMiddle, toTemp, array, from + middle, to, comparator);
		int index = from;
		int index1 = fromTemp;
		int index2 = tempMiddle;
		while (index1 < tempMiddle && index2 < toTemp) {
			if (comparator.compare(temp[index1], temp[index2]) <= 0)
				array[index++] = temp[index1++];
			else
				array[index++] = temp[index2++];
		}
		if (index1 != tempMiddle)
			System.arraycopy(temp, index1, array, index, tempMiddle - index1);
		if (index2 != toTemp)
			System.arraycopy(temp, index2, array, index, toTemp - index2);
	}

	public static int[] order(final int[] array) {
		return sort(createOrder(array.length), new IntComparator() {
			public int compare(Integer first, Integer second) {
				if (array[first] < array[second])
					return -1;
				if (array[first] > array[second])
					return 1;
				return 0;
			}
		});
	}

	public static int[] order(final long[] array) {
		return sort(createOrder(array.length), new IntComparator() {
			public int compare(Integer first, Integer second) {
				if (array[first] < array[second])
					return -1;
				if (array[first] > array[second])
					return 1;
				return 0;
			}
		});
	}

	public static int[] unique(int[] array) {
		return unique(array, 0, array.length);
	}

	public static int[] unique(int[] array, int from, int to) {
		if (from == to)
			return new int[0];
		int count = 1;
		for (int i = from + 1; i < to; i++) {
			if (array[i] != array[i - 1])
				count++;
		}
		int[] result = new int[count];
		result[0] = array[from];
		int index = 1;
		for (int i = from + 1; i < to; i++) {
			if (array[i] != array[i - 1])
				result[index++] = array[i];
		}
		return result;
	}

	public static char[] unique(char[] array) {
		return unique(array, 0, array.length);
	}

	public static char[] unique(char[] array, int from, int to) {
		if (from == to)
			return new char[0];
		int count = 1;
		for (int i = from + 1; i < to; i++) {
			if (array[i] != array[i - 1])
				count++;
		}
		char[] result = new char[count];
		result[0] = array[from];
		int index = 1;
		for (int i = from + 1; i < to; i++) {
			if (array[i] != array[i - 1])
				result[index++] = array[i];
		}
		return result;
	}

	public static int maxElement(int[] array) {
		return maxElement(array, 0, array.length);
	}

	public static int maxElement(int[] array, int from, int to) {
		int result = Integer.MIN_VALUE;
		for (int i = from; i < to; i++)
			result = Math.max(result, array[i]);
		return result;
	}

	public static int[] order(final double[] array) {
		return sort(createOrder(array.length), new IntComparator() {
			public int compare(Integer first, Integer second) {
				return Double.compare(array[first], array[second]);
			}
		});
	}

	public static int[] reversePermutation(int[] permutation) {
		int[] result = new int[permutation.length];
		for (int i = 0; i < permutation.length; i++)
			result[permutation[i]] = i;
		return result;
	}

	public static void reverse(int[] array) {
		for (int i = 0, j = array.length - 1; i < j; i++, j--) {
			int temp = array[i];
			array[i] = array[j];
			array[j] = temp;
		}
	}

	public static void reverse(char[] array) {
		for (int i = 0, j = array.length - 1; i < j; i++, j--) {
			char temp = array[i];
			array[i] = array[j];
			array[j] = temp;
		}
	}

	private static long maxElement(long[] array, int from, int to) {
		long result = Long.MIN_VALUE;
		for (int i = from; i < to; i++)
			result = Math.max(result, array[i]);
		return result;
	}

	public static int minPosition(int[] array) {
		return minPosition(array, 0, array.length);
	}

	public static int maxPosition(int[] array) {
		return maxPosition(array, 0, array.length);
	}

	public static int minPosition(int[] array, int from, int to) {
		if (from >= to)
			return -1;
		int min = array[from];
		int result = from;
		for (int i = from + 1; i < to; i++) {
			if (array[i] < min) {
				min = array[i];
				result = i;
			}
		}
		return result;
	}

	public static int maxPosition(int[] array, int from, int to) {
		if (from >= to)
			return -1;
		int max = array[from];
		int result = from;
		for (int i = from + 1; i < to; i++) {
			if (array[i] > max) {
				max = array[i];
				result = i;
			}
		}
		return result;
	}

	public static int[] multiplyPermutations(int[] first, int[] second) {
		int count = first.length;
		int[] result = new int[count];
		for (int i = 0; i < count; i++) {
			result[i] = first[second[i]];
		}
		return result;
	}

	public static int[] compress(int[]... arrays) {
		int totalLength = 0;
		for (int[] array : arrays)
			totalLength += array.length;
		int[] all = new int[totalLength];
		int delta = 0;
		for (int[] array : arrays) {
			System.arraycopy(array, 0, all, delta, array.length);
			delta += array.length;
		}
		sort(all, IntComparator.DEFAULT);
		all = unique(all);
		for (int[] array : arrays) {
			for (int i = 0; i < array.length; i++)
				array[i] = Arrays.binarySearch(all, array[i]);
		}
		return all;
	}

	public static int minElement(int[] array) {
		return array[minPosition(array)];
	}

	public static long[] partialSums(int[] array) {
		long[] result = new long[array.length + 1];
		for (int i = 0; i < array.length; i++)
			result[i + 1] = result[i] + array[i];
		return result;
	}

	public static void orderBy(int[] base, int[]... arrays) {
		int[] order = ArrayUtils.order(base);
		order(order, base);
		for (int[] array : arrays)
			order(order, array);
	}

	public static void orderBy(long[] base, long[]... arrays) {
		int[] order = ArrayUtils.order(base);
		order(order, base);
		for (long[] array : arrays)
			order(order, array);
	}

	public static void order(int[] order, int[] array) {
		ensureCapacityInt(order.length);
		for (int i = 0; i < order.length; i++)
			tempInt[i] = array[order[i]];
		System.arraycopy(tempInt, 0, array, 0, array.length);
	}

	public static void order(int[] order, long[] array) {
		ensureCapacityLong(order.length);
		for (int i = 0; i < order.length; i++)
			tempLong[i] = array[order[i]];
		System.arraycopy(tempLong, 0, array, 0, array.length);
	}

	public static long[] asLong(int[] array) {
		long[] result = new long[array.length];
		for (int i = 0; i < array.length; i++)
			result[i] = array[i];
		return result;
	}

	public static int count(int[] array, int value) {
		int result = 0;
		for (int i : array) {
			if (i == value)
				result++;
		}
		return result;
	}

	public static int count(char[] array, char value) {
		int result = 0;
		for (char i : array) {
			if (i == value)
				result++;
		}
		return result;
	}

	public static int count(boolean[] array, boolean value) {
		int result = 0;
		for (boolean i : array) {
			if (i == value)
				result++;
		}
		return result;
	}

	public static int[] merge(int[] first, int[] second) {
		int[] result = new int[first.length + second.length];
		int firstIndex = 0;
		int secondIndex = 0;
		int index = 0;
		while (firstIndex < first.length && secondIndex < second.length) {
			if (first[firstIndex] < second[secondIndex])
				result[index++] = first[firstIndex++];
			else
				result[index++] = second[secondIndex++];
		}
		System.arraycopy(first, firstIndex, result, index, first.length
				- firstIndex);
		System.arraycopy(second, secondIndex, result, index, second.length
				- secondIndex);
		return result;
	}

	public static boolean nextPermutation(int[] array) {
		return nextPermutation(array, IntComparator.DEFAULT);
	}

	private static boolean nextPermutation(int[] array, IntComparator comparator) {
		int size = array.length;
		int last = array[size - 1];
		for (int i = size - 2; i >= 0; i--) {
			int current = array[i];
			if (comparator.compare(last, current) > 0) {
				for (int j = size - 1; j > i; j--) {
					if (comparator.compare(array[j], current) > 0) {
						swap(array, i, j);
						inPlaceReverse(array, i + 1, size);
						return true;
					}
				}
			}
			last = current;
		}
		return false;
	}

	private static void inPlaceReverse(int[] array, int first, int second) {
		for (int i = first, j = second - 1; i < j; i++, j--)
			swap(array, i, j);
	}

	private static void swap(int[] array, int first, int second) {
		if (first == second)
			return;
		int temp = array[first];
		array[first] = array[second];
		array[second] = temp;
	}

	public static <V> void reverse(V[] array) {
		for (int i = 0, j = array.length - 1; i < j; i++, j--) {
			V temp = array[i];
			array[i] = array[j];
			array[j] = temp;
		}
	}

	public static IntComparator compareBy(final int[]... arrays) {
		return new IntComparator() {
			public int compare(Integer first, Integer second) {
				for (int[] array : arrays) {
					if (array[first] != array[second])
						return Integer.compare(array[first], array[second]);
				}
				return 0;
			}
		};
	}

	public static long minElement(long[] array) {
		return array[minPosition(array)];
	}

	public static long maxElement(long[] array) {
		return array[maxPosition(array)];
	}

	public static int minPosition(long[] array) {
		return minPosition(array, 0, array.length);
	}

	public static int maxPosition(long[] array) {
		return maxPosition(array, 0, array.length);
	}

	public static int minPosition(long[] array, int from, int to) {
		if (from >= to)
			return -1;
		long min = array[from];
		int result = from;
		for (int i = from + 1; i < to; i++) {
			if (array[i] < min) {
				min = array[i];
				result = i;
			}
		}
		return result;
	}

	public static int maxPosition(long[] array, int from, int to) {
		if (from >= to)
			return -1;
		long max = array[from];
		int result = from;
		for (int i = from + 1; i < to; i++) {
			if (array[i] > max) {
				max = array[i];
				result = i;
			}
		}
		return result;
	}

	public static int[] createArray(int count, int value) {
		int[] array = new int[count];
		Arrays.fill(array, value);
		return array;
	}

	public static long[] createArray(int count, long value) {
		long[] array = new long[count];
		Arrays.fill(array, value);
		return array;
	}

	public static double[] createArray(int count, double value) {
		double[] array = new double[count];
		Arrays.fill(array, value);
		return array;
	}

	public static boolean[] createArray(int count, boolean value) {
		boolean[] array = new boolean[count];
		Arrays.fill(array, value);
		return array;
	}

	public static char[] createArray(int count, char value) {
		char[] array = new char[count];
		Arrays.fill(array, value);
		return array;
	}

	public static <T> T[] createArray(int count, T value) {
		@SuppressWarnings("unchecked")
		T[] array = (T[]) Array.newInstance(value.getClass(), count);
		Arrays.fill(array, value);
		return array;
	}
}

class Graph {
	public static final int REMOVED_BIT = 0;

	protected int vertexCount;
	protected int edgeCount;

	private int[] firstOutbound;
	private int[] firstInbound;

	private Edge[] edges;
	private int[] nextInbound;
	private int[] nextOutbound;
	private int[] from;
	private int[] to;
	private long[] weight;
	private long[] capacity;
	private int[] reverseEdge;
	private int[] flags;

	public Graph(int vertexCount) {
		this(vertexCount, vertexCount);
	}

	public Graph(int vertexCount, int edgeCapacity) {
		this.vertexCount = vertexCount;
		firstOutbound = new int[vertexCount];
		Arrays.fill(firstOutbound, -1);

		from = new int[edgeCapacity];
		to = new int[edgeCapacity];
		nextOutbound = new int[edgeCapacity];
		flags = new int[edgeCapacity];
	}

	public static Graph createGraph(int vertexCount, int[] from, int[] to) {
		Graph graph = new Graph(vertexCount, from.length);
		for (int i = 0; i < from.length; i++)
			graph.addSimpleEdge(from[i], to[i]);
		return graph;
	}

	public static Graph createWeightedGraph(int vertexCount, int[] from,
			int[] to, long[] weight) {
		Graph graph = new Graph(vertexCount, from.length);
		for (int i = 0; i < from.length; i++)
			graph.addWeightedEdge(from[i], to[i], weight[i]);
		return graph;
	}

	public static Graph createFlowGraph(int vertexCount, int[] from, int[] to,
			long[] capacity) {
		Graph graph = new Graph(vertexCount, from.length * 2);
		for (int i = 0; i < from.length; i++)
			graph.addFlowEdge(from[i], to[i], capacity[i]);
		return graph;
	}

	public static Graph createFlowWeightedGraph(int vertexCount, int[] from,
			int[] to, long[] weight, long[] capacity) {
		Graph graph = new Graph(vertexCount, from.length * 2);
		for (int i = 0; i < from.length; i++)
			graph.addFlowWeightedEdge(from[i], to[i], weight[i], capacity[i]);
		return graph;
	}

	public static Graph createTree(int[] parent) {
		Graph graph = new Graph(parent.length + 1, parent.length);
		for (int i = 0; i < parent.length; i++)
			graph.addSimpleEdge(parent[i], i + 1);
		return graph;
	}

	public int addEdge(int fromID, int toID, long weight, long capacity,
			int reverseEdge) {
		ensureEdgeCapacity(edgeCount + 1);
		if (firstOutbound[fromID] != -1)
			nextOutbound[edgeCount] = firstOutbound[fromID];
		else
			nextOutbound[edgeCount] = -1;
		firstOutbound[fromID] = edgeCount;
		if (firstInbound != null) {
			if (firstInbound[toID] != -1)
				nextInbound[edgeCount] = firstInbound[toID];
			else
				nextInbound[edgeCount] = -1;
			firstInbound[toID] = edgeCount;
		}
		this.from[edgeCount] = fromID;
		this.to[edgeCount] = toID;
		if (capacity != 0) {
			if (this.capacity == null)
				this.capacity = new long[from.length];
			this.capacity[edgeCount] = capacity;
		}
		if (weight != 0) {
			if (this.weight == null)
				this.weight = new long[from.length];
			this.weight[edgeCount] = weight;
		}
		if (reverseEdge != -1) {
			if (this.reverseEdge == null) {
				this.reverseEdge = new int[from.length];
				Arrays.fill(this.reverseEdge, 0, edgeCount, -1);
			}
			this.reverseEdge[edgeCount] = reverseEdge;
		}
		if (edges != null)
			edges[edgeCount] = createEdge(edgeCount);
		return edgeCount++;
	}

	protected final GraphEdge createEdge(int id) {
		return new GraphEdge(id);
	}

	public final int addFlowWeightedEdge(int from, int to, long weight,
			long capacity) {
		if (capacity == 0) {
			return addEdge(from, to, weight, 0, -1);
		} else {
			int lastEdgeCount = edgeCount;
			addEdge(to, from, -weight, 0, lastEdgeCount + entriesPerEdge());
			return addEdge(from, to, weight, capacity, lastEdgeCount);
		}
	}

	protected int entriesPerEdge() {
		return 1;
	}

	public final int addFlowEdge(int from, int to, long capacity) {
		return addFlowWeightedEdge(from, to, 0, capacity);
	}

	public final int addWeightedEdge(int from, int to, long weight) {
		return addFlowWeightedEdge(from, to, weight, 0);
	}

	public final int addSimpleEdge(int from, int to) {
		return addWeightedEdge(from, to, 0);
	}

	public final int vertexCount() {
		return vertexCount;
	}

	public final int edgeCount() {
		return edgeCount;
	}

	protected final int edgeCapacity() {
		return from.length;
	}

	public final Edge edge(int id) {
		initEdges();
		return edges[id];
	}

	public final int firstOutbound(int vertex) {
		int id = firstOutbound[vertex];
		while (id != -1 && isRemoved(id))
			id = nextOutbound[id];
		return id;
	}

	public final int nextOutbound(int id) {
		id = nextOutbound[id];
		while (id != -1 && isRemoved(id))
			id = nextOutbound[id];
		return id;
	}

	public final int firstInbound(int vertex) {
		initInbound();
		int id = firstInbound[vertex];
		while (id != -1 && isRemoved(id))
			id = nextInbound[id];
		return id;
	}

	public final int nextInbound(int id) {
		initInbound();
		id = nextInbound[id];
		while (id != -1 && isRemoved(id))
			id = nextInbound[id];
		return id;
	}

	public final int source(int id) {
		return from[id];
	}

	public final int destination(int id) {
		return to[id];
	}

	public final long weight(int id) {
		if (weight == null)
			return 0;
		return weight[id];
	}

	public final long capacity(int id) {
		if (capacity == null)
			return 0;
		return capacity[id];
	}

	public final long flow(int id) {
		if (reverseEdge == null)
			return 0;
		return capacity[reverseEdge[id]];
	}

	public final void pushFlow(int id, long flow) {
		if (flow == 0)
			return;
		if (flow > 0) {
			if (capacity(id) < flow)
				throw new IllegalArgumentException("Not enough capacity");
		} else {
			if (flow(id) < -flow)
				throw new IllegalArgumentException("Not enough capacity");
		}
		capacity[id] -= flow;
		capacity[reverseEdge[id]] += flow;
	}

	public int transposed(int id) {
		return -1;
	}

	public final int reverse(int id) {
		if (reverseEdge == null)
			return -1;
		return reverseEdge[id];
	}

	public final void addVertices(int count) {
		ensureVertexCapacity(vertexCount + count);
		Arrays.fill(firstOutbound, vertexCount, vertexCount + count, -1);
		if (firstInbound != null)
			Arrays.fill(firstInbound, vertexCount, vertexCount + count, -1);
		vertexCount += count;
	}

	protected final void initEdges() {
		if (edges == null) {
			edges = new Edge[from.length];
			for (int i = 0; i < edgeCount; i++)
				edges[i] = createEdge(i);
		}
	}

	public final void removeVertex(int vertex) {
		int id = firstOutbound[vertex];
		while (id != -1) {
			removeEdge(id);
			id = nextOutbound[id];
		}
		initInbound();
		id = firstInbound[vertex];
		while (id != -1) {
			removeEdge(id);
			id = nextInbound[id];
		}
	}

	private void initInbound() {
		if (firstInbound == null) {
			firstInbound = new int[firstOutbound.length];
			Arrays.fill(firstInbound, 0, vertexCount, -1);
			nextInbound = new int[from.length];
			for (int i = 0; i < edgeCount; i++) {
				nextInbound[i] = firstInbound[to[i]];
				firstInbound[to[i]] = i;
			}
		}
	}

	public final boolean flag(int id, int bit) {
		return (flags[id] >> bit & 1) != 0;
	}

	public final void setFlag(int id, int bit) {
		flags[id] |= 1 << bit;
	}

	public final void removeFlag(int id, int bit) {
		flags[id] &= -1 - (1 << bit);
	}

	public final void removeEdge(int id) {
		setFlag(id, REMOVED_BIT);
	}

	public final void restoreEdge(int id) {
		removeFlag(id, REMOVED_BIT);
	}

	public final boolean isRemoved(int id) {
		return flag(id, REMOVED_BIT);
	}

	public final Iterable<Edge> outbound(final int id) {
		initEdges();
		return new Iterable<Edge>() {
			public Iterator<Edge> iterator() {
				return new EdgeIterator(id, firstOutbound, nextOutbound);
			}
		};
	}

	public final Iterable<Edge> inbound(final int id) {
		initEdges();
		initInbound();
		return new Iterable<Edge>() {
			public Iterator<Edge> iterator() {
				return new EdgeIterator(id, firstInbound, nextInbound);
			}
		};
	}

	protected void ensureEdgeCapacity(int size) {
		if (from.length < size) {
			int newSize = Math.max(size, 2 * from.length);
			if (edges != null)
				edges = resize(edges, newSize);
			from = resize(from, newSize);
			to = resize(to, newSize);
			nextOutbound = resize(nextOutbound, newSize);
			if (nextInbound != null)
				nextInbound = resize(nextInbound, newSize);
			if (weight != null)
				weight = resize(weight, newSize);
			if (capacity != null)
				capacity = resize(capacity, newSize);
			if (reverseEdge != null)
				reverseEdge = resize(reverseEdge, newSize);
			flags = resize(flags, newSize);
		}
	}

	private void ensureVertexCapacity(int size) {
		if (firstOutbound.length < size) {
			int newSize = Math.max(size, 2 * from.length);
			firstOutbound = resize(firstOutbound, newSize);
			if (firstInbound != null)
				firstInbound = resize(firstInbound, newSize);
		}
	}

	protected final int[] resize(int[] array, int size) {
		int[] newArray = new int[size];
		System.arraycopy(array, 0, newArray, 0, array.length);
		return newArray;
	}

	private long[] resize(long[] array, int size) {
		long[] newArray = new long[size];
		System.arraycopy(array, 0, newArray, 0, array.length);
		return newArray;
	}

	private Edge[] resize(Edge[] array, int size) {
		Edge[] newArray = new Edge[size];
		System.arraycopy(array, 0, newArray, 0, array.length);
		return newArray;
	}

	public final boolean isSparse() {
		return vertexCount == 0 || edgeCount * 20 / vertexCount <= vertexCount;
	}

	protected class GraphEdge implements Edge {
		protected int id;

		protected GraphEdge(int id) {
			this.id = id;
		}

		public int getSource() {
			return source(id);
		}

		public int getDestination() {
			return destination(id);
		}

		public long getWeight() {
			return weight(id);
		}

		public long getCapacity() {
			return capacity(id);
		}

		public long getFlow() {
			return flow(id);
		}

		public void pushFlow(long flow) {
			Graph.this.pushFlow(id, flow);
		}

		public boolean getFlag(int bit) {
			return flag(id, bit);
		}

		public void setFlag(int bit) {
			Graph.this.setFlag(id, bit);
		}

		public void removeFlag(int bit) {
			Graph.this.removeFlag(id, bit);
		}

		public int getTransposedID() {
			return transposed(id);
		}

		public Edge getTransposedEdge() {
			int reverseID = getTransposedID();
			if (reverseID == -1)
				return null;
			initEdges();
			return edge(reverseID);
		}

		public int getReverseID() {
			return reverse(id);
		}

		public Edge getReverseEdge() {
			int reverseID = getReverseID();
			if (reverseID == -1)
				return null;
			initEdges();
			return edge(reverseID);
		}

		public int getID() {
			return id;
		}

		public void remove() {
			removeEdge(id);
		}

		public void restore() {
			restoreEdge(id);
		}
	}

	public class EdgeIterator implements Iterator<Edge> {
		private int edgeID;
		private final int[] next;
		private int lastID = -1;

		public EdgeIterator(int id, int[] first, int[] next) {
			this.next = next;
			edgeID = nextEdge(first[id]);
		}

		private int nextEdge(int id) {
			while (id != -1 && isRemoved(id))
				id = next[id];
			return id;
		}

		public boolean hasNext() {
			return edgeID != -1;
		}

		public Edge next() {
			if (edgeID == -1)
				throw new NoSuchElementException();
			lastID = edgeID;
			edgeID = nextEdge(next[lastID]);
			return edges[lastID];
		}

		public void remove() {
			if (lastID == -1)
				throw new IllegalStateException();
			removeEdge(lastID);
			lastID = -1;
		}
	}

}

class MiscUtils {
	public static final int[] DX4 = { 1, 0, -1, 0 };
	public static final int[] DY4 = { 0, -1, 0, 1 };
	public static final int[] DX8 = { 1, 1, 1, 0, -1, -1, -1, 0 };
	public static final int[] DY8 = { -1, 0, 1, 1, 1, 0, -1, -1 };
	public static final int[] DX_KNIGHT = { 2, 1, -1, -2, -2, -1, 1, 2 };
	public static final int[] DY_KNIGHT = { 1, 2, 2, 1, -1, -2, -2, -1 };

	private static final String[] ROMAN_TOKENS = { "M", "CM", "D", "CD", "C",
			"XC", "L", "XL", "X", "IX", "V", "IV", "I" };
	private static final int[] ROMAN_VALUES = { 1000, 900, 500, 400, 100, 90,
			50, 40, 10, 9, 5, 4, 1 };

	public static long josephProblem(long n, int k) {
		if (n == 1)
			return 0;
		if (k == 1)
			return n - 1;
		if (k > n)
			return (josephProblem(n - 1, k) + k) % n;
		long count = n / k;
		long result = josephProblem(n - count, k);
		result -= n % k;
		if (result < 0)
			result += n;
		else
			result += result / (k - 1);
		return result;
	}

	public static boolean isValidCell(int row, int column, int rowCount,
			int columnCount) {
		return row >= 0 && row < rowCount && column >= 0
				&& column < columnCount;
	}

	public static List<Integer> getPath(int[] last, int destination) {
		List<Integer> path = new ArrayList<Integer>();
		while (destination != -1) {
			path.add(destination);
			destination = last[destination];
		}
		inPlaceReverse(path);
		return path;
	}

	private static void inPlaceReverse(List<Integer> list) {
		for (int i = 0, j = list.size() - 1; i < j; i++, j--)
			swap(list, i, j);
	}

	private static void swap(List<Integer> list, int first, int second) {
		if (first == second)
			return;
		int temp = list.get(first);
		list.set(first, list.get(second));
		list.set(second, temp);
	}

	public static List<Integer> getPath(int[][] lastIndex,
			int[][] lastPathNumber, int destination, int pathNumber) {
		List<Integer> path = new ArrayList<Integer>();
		while (destination != -1 || pathNumber != 0) {
			path.add(destination);
			int nextDestination = lastIndex[destination][pathNumber];
			pathNumber = lastPathNumber[destination][pathNumber];
			destination = nextDestination;
		}
		inPlaceReverse(path);
		return path;
	}

	public static long maximalRectangleSum(long[][] array) {
		int n = array.length;
		int m = array[0].length;
		long[][] partialSums = new long[n + 1][m + 1];
		for (int i = 0; i < n; i++) {
			long rowSum = 0;
			for (int j = 0; j < m; j++) {
				rowSum += array[i][j];
				partialSums[i + 1][j + 1] = partialSums[i][j + 1] + rowSum;
			}
		}
		long result = Long.MIN_VALUE;
		for (int i = 0; i < m; i++) {
			for (int j = i; j < m; j++) {
				long minPartialSum = 0;
				for (int k = 1; k <= n; k++) {
					long current = partialSums[k][j + 1] - partialSums[k][i];
					result = Math.max(result, current - minPartialSum);
					minPartialSum = Math.min(minPartialSum, current);
				}
			}
		}
		return result;
	}

	public static int parseIP(String ip) {
		String[] components = ip.split("[.]");
		int result = 0;
		for (int i = 0; i < 4; i++)
			result += (1 << (24 - 8 * i)) * Integer.parseInt(components[i]);
		return result;
	}

	public static String buildIP(int mask) {
		StringBuilder result = new StringBuilder();
		for (int i = 0; i < 4; i++) {
			if (i != 0)
				result.append('.');
			result.append(mask >> (24 - 8 * i) & 255);
		}
		return result.toString();
	}

	public static long binarySearch(long from, long to,
			Function<Long, Boolean> function) {
		while (from < to) {
			long argument = from + (to - from) / 2;
			if (function.value(argument))
				to = argument;
			else
				from = argument + 1;
		}
		return from;
	}

	public static <T> boolean equals(T first, T second) {
		return first == null && second == null || first != null
				&& first.equals(second);
	}

	public static boolean isVowel(char ch) {
		ch = Character.toUpperCase(ch);
		return ch == 'A' || ch == 'E' || ch == 'I' || ch == 'O' || ch == 'U'
				|| ch == 'Y';
	}

	public static boolean isStrictVowel(char ch) {
		ch = Character.toUpperCase(ch);
		return ch == 'A' || ch == 'E' || ch == 'I' || ch == 'O' || ch == 'U';
	}

	public static String convertToRoman(int number) {
		StringBuilder result = new StringBuilder();
		for (int i = 0; i < ROMAN_TOKENS.length; i++) {
			while (number >= ROMAN_VALUES[i]) {
				number -= ROMAN_VALUES[i];
				result.append(ROMAN_TOKENS[i]);
			}
		}
		return result.toString();
	}

	public static int convertFromRoman(String number) {
		int result = 0;
		for (int i = 0; i < ROMAN_TOKENS.length; i++) {
			while (number.startsWith(ROMAN_TOKENS[i])) {
				number = number.substring(ROMAN_TOKENS[i].length());
				result += ROMAN_VALUES[i];
			}
		}
		return result;
	}

	public static int distance(int x1, int y1, int x2, int y2) {
		int dx = x1 - x2;
		int dy = y1 - y2;
		return dx * dx + dy * dy;
	}

	public static <T extends Comparable<T>> T min(T first, T second) {
		if (first.compareTo(second) <= 0)
			return first;
		return second;
	}

	public static <T extends Comparable<T>> T max(T first, T second) {
		if (first.compareTo(second) <= 0)
			return second;
		return first;
	}

	public static void decreaseByOne(int[]... arrays) {
		for (int[] array : arrays) {
			for (int i = 0; i < array.length; i++)
				array[i]--;
		}
	}

	public static int[] getIntArray(String s) {
		String[] tokens = s.split(" ");
		int[] result = new int[tokens.length];
		for (int i = 0; i < result.length; i++)
			result[i] = Integer.parseInt(tokens[i]);
		return result;
	}
}

class IntegerUtils {
	public static long gcd(long a, long b) {
		a = Math.abs(a);
		b = Math.abs(b);
		while (b != 0) {
			long temp = a % b;
			a = b;
			b = temp;
		}
		return a;
	}

	public static int gcd(int a, int b) {
		a = Math.abs(a);
		b = Math.abs(b);
		while (b != 0) {
			int temp = a % b;
			a = b;
			b = temp;
		}
		return a;
	}

	public static int[] generatePrimes(int upTo) {
		int[] isPrime = generateBitPrimalityTable(upTo);
		List<Integer> primes = new ArrayList<Integer>();
		for (int i = 0; i < upTo; i++) {
			if ((isPrime[i >> 5] >>> (i & 31) & 1) == 1)
				primes.add(i);
		}
		int[] array = new int[primes.size()];
		for (int i = 0; i < array.length; i++)
			array[i] = primes.get(i);
		return array;
	}

	public static boolean[] generatePrimalityTable(int upTo) {
		boolean[] isPrime = new boolean[upTo];
		if (upTo < 2)
			return isPrime;
		Arrays.fill(isPrime, true);
		isPrime[0] = isPrime[1] = false;
		for (int i = 2; i * i < upTo; i++) {
			if (isPrime[i]) {
				for (int j = i * i; j < upTo; j += i)
					isPrime[j] = false;
			}
		}
		return isPrime;
	}

	public static int[] generateBitPrimalityTable(int upTo) {
		int[] isPrime = new int[(upTo + 31) >> 5];
		if (upTo < 2)
			return isPrime;
		Arrays.fill(isPrime, -1);
		isPrime[0] &= -4;
		for (int i = 2; i * i < upTo; i++) {
			if ((isPrime[i >> 5] >>> (i & 31) & 1) == 1) {
				for (int j = i * i; j < upTo; j += i)
					isPrime[j >> 5] &= -1 - (1 << (j & 31));
			}
		}
		return isPrime;
	}

	public static int[] generateDivisorTable(int upTo) {
		int[] divisor = new int[upTo];
		for (int i = 1; i < upTo; i++)
			divisor[i] = i;
		for (int i = 2; i * i < upTo; i++) {
			if (divisor[i] == i) {
				for (int j = i * i; j < upTo; j += i)
					divisor[j] = i;
			}
		}
		return divisor;
	}

	public static long powerInFactorial(long n, long p) {
		long result = 0;
		while (n != 0) {
			result += n /= p;
		}
		return result;
	}

	public static int sumDigits(CharSequence number) {
		int result = 0;
		for (int i = number.length() - 1; i >= 0; i--)
			result += digitValue(number.charAt(i));
		return result;
	}

	public static int digitValue(char digit) {
		if (Character.isDigit(digit))
			return digit - '0';
		if (Character.isUpperCase(digit))
			return digit + 10 - 'A';
		return digit + 10 - 'a';
	}

	public static int longCompare(long a, long b) {
		if (a < b)
			return -1;
		if (a > b)
			return 1;
		return 0;
	}

	public static long[][] generateBinomialCoefficients(int n) {
		long[][] result = new long[n + 1][n + 1];
		for (int i = 0; i <= n; i++) {
			result[i][0] = 1;
			for (int j = 1; j <= i; j++)
				result[i][j] = result[i - 1][j - 1] + result[i - 1][j];
		}
		return result;
	}

	public static long[][] generateBinomialCoefficients(int n, long module) {
		long[][] result = new long[n + 1][n + 1];
		if (module == 1)
			return result;
		for (int i = 0; i <= n; i++) {
			result[i][0] = 1;
			for (int j = 1; j <= i; j++) {
				result[i][j] = result[i - 1][j - 1] + result[i - 1][j];
				if (result[i][j] >= module)
					result[i][j] -= module;
			}
		}
		return result;
	}

	public static long[] generateBinomialRow(int n, long module) {
		long[] result = generateReverse(n + 1, module);
		result[0] = 1;
		for (int i = 1; i <= n; i++)
			result[i] = result[i - 1] * (n - i + 1) % module * result[i]
					% module;
		return result;
	}

	public static int[] representationInBase(long number, int base) {
		long basePower = base;
		int exponent = 1;
		while (number >= basePower) {
			basePower *= base;
			exponent++;
		}
		int[] representation = new int[exponent];
		for (int i = 0; i < exponent; i++) {
			basePower /= base;
			representation[i] = (int) (number / basePower);
			number %= basePower;
		}
		return representation;
	}

	public static int trueDivide(int a, int b) {
		return (a - trueMod(a, b)) / b;
	}

	public static long trueDivide(long a, long b) {
		return (a - trueMod(a, b)) / b;
	}

	public static int trueMod(int a, int b) {
		a %= b;
		a += b;
		a %= b;
		return a;
	}

	public static long trueMod(long a, long b) {
		a %= b;
		a += b;
		a %= b;
		return a;
	}

	public static long factorial(int n) {
		long result = 1;
		for (int i = 2; i <= n; i++)
			result *= i;
		return result;
	}

	public static long factorial(int n, long mod) {
		long result = 1;
		for (int i = 2; i <= n; i++)
			result = result * i % mod;
		return result % mod;
	}

	public static List<Pair<Long, Integer>> factorize(long number) {
		List<Pair<Long, Integer>> result = new ArrayList<Pair<Long, Integer>>();
		for (long i = 2; i * i <= number; i++) {
			if (number % i == 0) {
				int power = 0;
				do {
					power++;
					number /= i;
				} while (number % i == 0);
				result.add(Pair.makePair(i, power));
			}
		}
		if (number != 1)
			result.add(Pair.makePair(number, 1));
		return result;
	}

	public static List<Long> getDivisors(long number) {
		List<Pair<Long, Integer>> primeDivisors = factorize(number);
		return getDivisorsImpl(primeDivisors, 0, 1, new ArrayList<Long>());
	}

	private static List<Long> getDivisorsImpl(
			List<Pair<Long, Integer>> primeDivisors, int index, long current,
			List<Long> result) {
		if (index == primeDivisors.size()) {
			result.add(current);
			return result;
		}
		long p = primeDivisors.get(index).first;
		int power = primeDivisors.get(index).second;
		for (int i = 0; i <= power; i++) {
			getDivisorsImpl(primeDivisors, index + 1, current, result);
			current *= p;
		}
		return result;
	}

	public static long power(long base, long exponent) {
		if (exponent == 0)
			return 1;
		long result = power(base, exponent >> 1);
		result = result * result;
		if ((exponent & 1) != 0)
			result = result * base;
		return result;
	}

	public static long power(long base, long exponent, long mod) {
		if (base >= mod)
			base %= mod;
		if (exponent == 0)
			return 1 % mod;
		long result = power(base, exponent >> 1, mod);
		result = result * result % mod;
		if ((exponent & 1) != 0)
			result = result * base % mod;
		return result;
	}

	public static long lcm(long a, long b) {
		return a / gcd(a, b) * b;
	}

	public static long[] generateFibonacci(long upTo) {
		int count = 0;
		long last = 0;
		long current = 1;
		while (current <= upTo) {
			long next = last + current;
			last = current;
			current = next;
			count++;
		}
		return generateFibonacci(count, -1);
	}

	public static long[] generateFibonacci(int count, long module) {
		long[] result = new long[count];
		if (module == -1) {
			if (count != 0)
				result[0] = 1;
			if (count > 1)
				result[1] = 1;
			for (int i = 2; i < count; i++)
				result[i] = result[i - 1] + result[i - 2];
		} else {
			if (count != 0)
				result[0] = 1 % module;
			if (count > 1)
				result[1] = 1 % module;
			for (int i = 2; i < count; i++)
				result[i] = (result[i - 1] + result[i - 2]) % module;
		}
		return result;
	}

	public static long[] generateHappy(int digits) {
		long[] happy = new long[(1 << (digits + 1)) - 2];
		happy[0] = 4;
		happy[1] = 7;
		int first = 0;
		int last = 2;
		for (int i = 2; i <= digits; i++) {
			for (int j = 0; j < last - first; j++) {
				happy[last + 2 * j] = 10 * happy[first + j] + 4;
				happy[last + 2 * j + 1] = 10 * happy[first + j] + 7;
			}
			int next = last + 2 * (last - first);
			first = last;
			last = next;
		}
		return happy;
	}

	public static long[] generateFactorial(int count, long module) {
		long[] result = new long[count];
		if (module == -1) {
			if (count != 0)
				result[0] = 1;
			for (int i = 1; i < count; i++)
				result[i] = result[i - 1] * i;
		} else {
			if (count != 0)
				result[0] = 1 % module;
			for (int i = 1; i < count; i++)
				result[i] = (result[i - 1] * i) % module;
		}
		return result;
	}

	public static long reverse(long number, long module) {
		return power(number, module - 2, module);
	}

	public static boolean isPrime(long number) {
		if (number < 2)
			return false;
		for (long i = 2; i * i <= number; i++) {
			if (number % i == 0)
				return false;
		}
		return true;
	}

	public static long[] generateReverse(int upTo, long module) {
		long[] result = new long[upTo];
		if (upTo > 1)
			result[1] = 1;
		for (int i = 2; i < upTo; i++)
			result[i] = (module - module / i * result[((int) (module % i))]
					% module)
					% module;
		return result;
	}

	public static long[] generateReverseFactorials(int upTo, long module) {
		long[] result = generateReverse(upTo, module);
		if (upTo > 0)
			result[0] = 1;
		for (int i = 1; i < upTo; i++)
			result[i] = result[i] * result[i - 1] % module;
		return result;
	}

	public static long[] generatePowers(long base, int count, long mod) {
		long[] result = new long[count];
		if (count != 0)
			result[0] = 1 % mod;
		for (int i = 1; i < count; i++)
			result[i] = result[i - 1] * base % mod;
		return result;
	}

	public static long nextPrime(long from) {
		if (from <= 2)
			return 2;
		from += 1 - (from & 1);
		while (!isPrime(from))
			from += 2;
		return from;
	}

	public static long binomialCoefficient(int n, int m, long mod) {
		if (m < 0 || m > n)
			return 0;
		if (2 * m > n)
			m = n - m;
		long result = 1;
		for (int i = n - m + 1; i <= n; i++)
			result = result * i % mod;
		return result
				* BigInteger.valueOf(factorial(m, mod))
						.modInverse(BigInteger.valueOf(mod)).longValue() % mod;
	}

	public static boolean isSquare(long number) {
		long sqrt = Math.round(Math.sqrt(number));
		return sqrt * sqrt == number;
	}

	public static long findCommon(long aRemainder, long aMod, long bRemainder,
			long bMod) {
		long modGCD = gcd(aMod, bMod);
		long gcdRemainder = aRemainder % modGCD;
		if (gcdRemainder != bRemainder % modGCD)
			return -1;
		aMod /= modGCD;
		aRemainder /= modGCD;
		bMod /= modGCD;
		bRemainder /= modGCD;
		long aReverse = BigInteger.valueOf(aMod)
				.modInverse(BigInteger.valueOf(bMod)).longValue();
		long bReverse = BigInteger.valueOf(bMod)
				.modInverse(BigInteger.valueOf(aMod)).longValue();
		long mod = aMod * bMod;
		return (bReverse * aRemainder % mod * bMod + aReverse * bRemainder
				% mod * aMod)
				% mod * modGCD + gcdRemainder;
	}

	public static long[] generatePowers(long base, long maxValue) {
		if (maxValue <= 0)
			return new long[0];
		int size = 1;
		long current = 1;
		while (maxValue / base >= current) {
			current *= base;
			size++;
		}
		return generatePowers(base, size, Long.MAX_VALUE);
	}
}

interface IntComparator extends Comparator<Integer> {
	public static final IntComparator DEFAULT = new IntComparator() {
		public int compare(Integer first, Integer second) {
			if (first < second)
				return -1;
			if (first > second)
				return 1;
			return 0;
		}
	};

	public static final IntComparator REVERSE = new IntComparator() {
		public int compare(Integer first, Integer second) {
			if (first < second)
				return 1;
			if (first > second)
				return -1;
			return 0;
		}
	};

	public int compare(Integer first, Integer second);
}

class BidirectionalGraph extends Graph {
	public int[] transposedEdge;

	public BidirectionalGraph(int vertexCount) {
		this(vertexCount, vertexCount);
	}

	public BidirectionalGraph(int vertexCount, int edgeCapacity) {
		super(vertexCount, 2 * edgeCapacity);
		transposedEdge = new int[2 * edgeCapacity];
	}

	public static BidirectionalGraph createGraph(int vertexCount, int[] from,
			int[] to) {
		BidirectionalGraph graph = new BidirectionalGraph(vertexCount,
				from.length);
		for (int i = 0; i < from.length; i++)
			graph.addSimpleEdge(from[i], to[i]);
		return graph;
	}

	public static BidirectionalGraph createWeightedGraph(int vertexCount,
			int[] from, int[] to, long[] weight) {
		BidirectionalGraph graph = new BidirectionalGraph(vertexCount,
				from.length);
		for (int i = 0; i < from.length; i++)
			graph.addWeightedEdge(from[i], to[i], weight[i]);
		return graph;
	}

	public static BidirectionalGraph createFlowGraph(int vertexCount,
			int[] from, int[] to, long[] capacity) {
		BidirectionalGraph graph = new BidirectionalGraph(vertexCount,
				from.length * 2);
		for (int i = 0; i < from.length; i++)
			graph.addFlowEdge(from[i], to[i], capacity[i]);
		return graph;
	}

	public static BidirectionalGraph createFlowWeightedGraph(int vertexCount,
			int[] from, int[] to, long[] weight, long[] capacity) {
		BidirectionalGraph graph = new BidirectionalGraph(vertexCount,
				from.length * 2);
		for (int i = 0; i < from.length; i++)
			graph.addFlowWeightedEdge(from[i], to[i], weight[i], capacity[i]);
		return graph;
	}

	@Override
	public int addEdge(int fromID, int toID, long weight, long capacity,
			int reverseEdge) {
		int lastEdgeCount = edgeCount;
		super.addEdge(fromID, toID, weight, capacity, reverseEdge);
		super.addEdge(toID, fromID, weight, capacity, reverseEdge == -1 ? -1
				: reverseEdge + 1);
		this.transposedEdge[lastEdgeCount] = lastEdgeCount + 1;
		this.transposedEdge[lastEdgeCount + 1] = lastEdgeCount;
		return lastEdgeCount;
	}

	@Override
	protected int entriesPerEdge() {
		return 2;
	}

	@Override
	public final int transposed(int id) {
		return transposedEdge[id];
	}

	@Override
	protected void ensureEdgeCapacity(int size) {
		if (size > edgeCapacity()) {
			super.ensureEdgeCapacity(size);
			transposedEdge = resize(transposedEdge, edgeCapacity());
		}
	}
}

class Pair<U, V> implements Comparable<Pair<U, V>> {
	public final U first;
	public final V second;

	public static <U, V> Pair<U, V> makePair(U first, V second) {
		return new Pair<U, V>(first, second);
	}

	private Pair(U first, V second) {
		this.first = first;
		this.second = second;
	}

	@Override
	public boolean equals(Object o) {
		if (this == o)
			return true;
		if (o == null || getClass() != o.getClass())
			return false;

		Pair pair = (Pair) o;

		return !(first != null ? !first.equals(pair.first) : pair.first != null)
				&& !(second != null ? !second.equals(pair.second)
						: pair.second != null);

	}

	@Override
	public int hashCode() {
		int result = first != null ? first.hashCode() : 0;
		result = 31 * result + (second != null ? second.hashCode() : 0);
		return result;
	}

	public Pair<V, U> swap() {
		return makePair(second, first);
	}

	@Override
	public String toString() {
		return "(" + first + "," + second + ")";
	}

	@SuppressWarnings({ "unchecked" })
	public int compareTo(Pair<U, V> o) {
		int value = ((Comparable<U>) first).compareTo(o.first);
		if (value != 0)
			return value;
		return ((Comparable<V>) second).compareTo(o.second);
	}
}

interface Edge {
	public int getSource();

	public int getDestination();

	public long getWeight();

	public long getCapacity();

	public long getFlow();

	public void pushFlow(long flow);

	public boolean getFlag(int bit);

	public void setFlag(int bit);

	public void removeFlag(int bit);

	public int getTransposedID();

	public Edge getTransposedEdge();

	public int getReverseID();

	public Edge getReverseEdge();

	public int getID();

	public void remove();

	public void restore();
}

interface Function<A, V> {
	public abstract V value(A argument);
}