Om Nom really likes candies and doesn't like spiders as they frequently steal candies. One day Om Nom fancied a walk in a park. Unfortunately, the park has some spiders and Om Nom doesn't want to see them at all.

<image>

The park can be represented as a rectangular n × m field. The park has k spiders, each spider at time 0 is at some cell of the field. The spiders move all the time, and each spider always moves in one of the four directions (left, right, down, up). In a unit of time, a spider crawls from his cell to the side-adjacent cell in the corresponding direction. If there is no cell in the given direction, then the spider leaves the park. The spiders do not interfere with each other as they move. Specifically, one cell can have multiple spiders at the same time.

Om Nom isn't yet sure where to start his walk from but he definitely wants:

  * to start walking at time 0 at an upper row cell of the field (it is guaranteed that the cells in this row do not contain any spiders); 
  * to walk by moving down the field towards the lowest row (the walk ends when Om Nom leaves the boundaries of the park). 



We know that Om Nom moves by jumping. One jump takes one time unit and transports the little monster from his cell to either a side-adjacent cell on the lower row or outside the park boundaries.

Each time Om Nom lands in a cell he sees all the spiders that have come to that cell at this moment of time. Om Nom wants to choose the optimal cell to start the walk from. That's why he wonders: for each possible starting cell, how many spiders will he see during the walk if he starts from this cell? Help him and calculate the required value for each possible starting cell.

Input

The first line contains three integers n, m, k (2 ≤ n, m ≤ 2000; 0 ≤ k ≤ m(n - 1)). 

Each of the next n lines contains m characters — the description of the park. The characters in the i-th line describe the i-th row of the park field. If the character in the line equals ".", that means that the corresponding cell of the field is empty; otherwise, the character in the line will equal one of the four characters: "L" (meaning that this cell has a spider at time 0, moving left), "R" (a spider moving right), "U" (a spider moving up), "D" (a spider moving down). 

It is guaranteed that the first row doesn't contain any spiders. It is guaranteed that the description of the field contains no extra characters. It is guaranteed that at time 0 the field contains exactly k spiders.

Output

Print m integers: the j-th integer must show the number of spiders Om Nom will see if he starts his walk from the j-th cell of the first row. The cells in any row of the field are numbered from left to right.

Examples

Input

3 3 4
...
R.L
R.U


Output

0 2 2 

Input

2 2 2
..
RL


Output

1 1 

Input

2 2 2
..
LR


Output

0 0 

Input

3 4 8
....
RRLL
UUUU


Output

1 3 3 1 

Input

2 2 2
..
UU


Output

0 0 

Note

Consider the first sample. The notes below show how the spider arrangement changes on the field over time:
    
    
      
    ...        ...        ..U       ...  
    R.L   ->   .*U   ->   L.R   ->  ...  
    R.U        .R.        ..R       ...  
      
    

Character "*" represents a cell that contains two spiders at the same time.

  * If Om Nom starts from the first cell of the first row, he won't see any spiders. 
  * If he starts from the second cell, he will see two spiders at time 1. 
  * If he starts from the third cell, he will see two spiders: one at time 1, the other one at time 2. 
3 7 14
.......
LDUDLLD
DLRDDLD
 4 5 15
.....
DRRLR
ULDLD
DLRRL
 10 8 30
........
.L.LDRR.
D.LDLR.U
..RL.L..
UR.UL...
.D.....L
R..UDULL
........
DD.U...D
........
 5 6 20
......
.UURD.
LUD.RR
U.LDDD
DDLDDU
 5 7 19
.......
RDLLLRL
UUR..U.
.D.DLLL
..R..UU
 8 9 28
.........
.R.LDR.D.
....UULU.
R.D..DL.L
.R..DLUDU
R........
.URU...UU
.....D.L.
 10 10 47
..........
.R.LLL.R.L
R.ULUL.RUL
RD..RL.L..
RU.D....LU
..RRR..DL.
.DRUUD.ULD
...D...LDD
...UUU.U..
R.....D...
 2 2 0
..
..
 100 2 45
..
.D
U.
..
U.
..
..
..
U.
..
..
D.
U.
..
..
.D
DU
..
UD
..
..
..
..
..
..
D.
U.
..
..
D.
U.
..
..
..
U.
..
..
.D
..
..
.D
..
..
.D
.U
D.
..
.D
..
..
UD
..
U.
..
U.
..
UD
..
U.
..
U.
..
..
..
U.
..
..
D.
..
..
U.
..
U.
..
UU
..
U.
..
U.
..
..
..
..
..
..
..
..
..
.D
..
..
D.
U.
.D
..
..
U.
.D
U.
..
 2 100 59
....................................................................................................
.DR.D..DLLR.LDRR..L.LDRRRDLD.LDRR.LLR.R...DRLD.RRLL.L.D..R.LD.DL....LR.LR.DRLD.....L.D..RD...D.LL.R.
 5 5 10
.....
RU.D.
..DLL
.D...
RL..L

0 0 0 2 2 0 0  1 2 2 1 0  6 1 4 3 0 3 2 3  0 1 0 0 1 1  1 4 2 2 1 3 3  1 2 2 3 2 4 2 2 3  1 2 6 6 7 1 0 5 5 4  0 0  23 3  0 0 0 1 0 0 0 1 1 0 0 2 0 0 0 1 1 1 0 1 0 0 0 1 1 2 0 0 1 0 0 0 1 2 1 0 0 1 0 1 0 0 0 1 1 0 0 0 2 2 0 1 0 0 0 0 0 0 2 0 0 0 1 0 0 0 0 1 0 0 2 0 0 1 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 1  1 2 1 0 1 
import java.util.*;
import java.io.*;
import java.math.*;

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 TaskB();
		solver.solve(1, in, out);
		out.close();
	}
}

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

class TaskA implements Task {

	ArrayList<Pair<Integer, Integer>> arr1, arr2, arr;

	public void solve(int testNumber, InputReader in, OutputWriter out) {
		int n = in.readInt(), x = in.readInt(), i1 = 0, i2 = 0, ret = 0, ret2 = 0, x2 = 0;
		arr1 = new ArrayList<Pair<Integer, Integer>>();
		arr2 = new ArrayList<Pair<Integer, Integer>>();
		arr = new ArrayList<Pair<Integer, Integer>>();
		while (n-- > 0) {
			int t = in.readInt(), a = in.readInt(), b = in.readInt();
			if (t == 0)
				arr1.add(Pair.makePair(a, -b));
			else
				arr2.add(Pair.makePair(a, -b));
		}
		Collections.sort(arr1);
		Collections.sort(arr2);
		for (int i = 0; i < arr1.size() && i < arr2.size(); i++) {
			arr.add(arr1.get(i));
			arr.add(arr2.get(i));
		}
		if (arr1.size() > arr2.size())
			arr.add(arr1.get(arr2.size()));
		x2 = x;
		for (int i = 0; i < arr.size(); i++) {
			if (arr.get(i).first <= x2) {
				ret++;
				x2 -= arr.get(i).second;
			} else
				break;
		}
		x2 = x;
		arr.clear();
		for (int i = 0; i < arr1.size() && i < arr2.size(); i++) {
			arr.add(arr1.get(i));
			arr.add(arr2.get(i));
		}
		if (arr2.size() > arr1.size())
			arr.add(arr2.get(arr1.size()));
		for (int i = 0; i < arr.size(); i++) {
			if (arr.get(i).first <= x2) {
				ret2++;
				x2 -= arr.get(i).second;
			} else
				break;
		}
		out.printLine(MiscUtils.max(ret, ret2));
	}

}

class TaskB implements Task {

	char[][] tab;
	boolean[][] L, R, U, D;
	int[] ret;

	public void solve(int testNumber, InputReader in, OutputWriter out) {
		int n = in.readInt(), m = in.readInt(), k = in.readInt();
		tab = IOUtils.readTable(in, n, m);
		// L=new boolean[n][m];
		// R=new boolean[n][m];
		// U=new boolean[n][m];
		// D=new boolean[n][m];
		ret = new int[m];
		for (int i = 0; i < n; i++)
			for (int j = 0; j < m; j++) {
				if (tab[i][j] == 'U') {
					// U[i][j]=true;
					if (i % 2 == 0)
						ret[j]++;
				}
				if (tab[i][j] == 'D') {
					// D[i][j]=true;
				}
				if (tab[i][j] == 'L') {
					// L[i][j]=true;
					if (j - i < m && j - i >= 0)
						ret[j - i]++;
				}
				if (tab[i][j] == 'R') {
					// R[i][j]=true;
					if (j + i < m)
						ret[j + i]++;
				}
			}
		out.printLine(ret);
	}

}

class TaskC implements Task {

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

	}

}

class TaskD implements Task {

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

	}

}

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 Hopcroft {
	private final int NIL = 0;
	private final int INF = Integer.MAX_VALUE;
	private ArrayList<Integer>[] Adj;
	private int[] Pair;
	private int[] Dist;
	private int cx, cy;

	/** Function BFS **/
	public boolean BFS() {
		Queue<Integer> queue = new LinkedList<Integer>();
		for (int v = 1; v <= cx; ++v)
			if (Pair[v] == NIL) {
				Dist[v] = 0;
				queue.add(v);
			} else
				Dist[v] = INF;

		Dist[NIL] = INF;

		while (!queue.isEmpty()) {
			int v = queue.poll();
			if (Dist[v] < Dist[NIL])
				for (int u : Adj[v])
					if (Dist[Pair[u]] == INF) {
						Dist[Pair[u]] = Dist[v] + 1;
						queue.add(Pair[u]);
					}
		}
		return Dist[NIL] != INF;
	}

	/** Function DFS **/
	public boolean DFS(int v) {
		if (v != NIL) {
			for (int u : Adj[v])
				if (Dist[Pair[u]] == Dist[v] + 1)
					if (DFS(Pair[u])) {
						Pair[u] = v;
						Pair[v] = u;
						return true;
					}

			Dist[v] = INF;
			return false;
		}
		return true;
	}

	/** Function to get maximum matching **/
	public int HopcroftKarp() {
		Pair = new int[cx + cy + 1];
		Dist = new int[cx + cy + 1];
		int matching = 0;
		while (BFS())
			for (int v = 1; v <= cx; ++v)
				if (Pair[v] == NIL)
					if (DFS(v))
						matching = matching + 1;
		return matching;
	}

	/** Function to make graph with vertices x , y **/
	public void makeGraph(int[] x, int[] y, int E) {
		Adj = new ArrayList[cx + cy + 1];
		for (int i = 0; i < Adj.length; ++i)
			Adj[i] = new ArrayList<Integer>();
		/** adding edges **/
		for (int i = 0; i < E; ++i)
			addEdge(x[i] + 1, y[i] + 1);
	}

	/** Function to add a edge **/
	public void addEdge(int u, int v) {
		Adj[u].add(cx + v);
		Adj[cx + v].add(u);
	}

	/** Main Method **/
	public static void main(String[] args) {
		Scanner scan = new Scanner(System.in);
		System.out.println("Hopcroft Algorithm Test\n");
		/** Make an object of Hopcroft class **/
		Hopcroft hc = new Hopcroft();

		/** Accept number of edges **/
		System.out.println("Enter number of edges\n");
		int E = scan.nextInt();
		int[] x = new int[E];
		int[] y = new int[E];
		hc.cx = 0;
		hc.cy = 0;

		System.out.println("Enter " + E + " x, y coordinates ");
		for (int i = 0; i < E; i++) {
			x[i] = scan.nextInt();
			y[i] = scan.nextInt();
			hc.cx = Math.max(hc.cx, x[i]);
			hc.cy = Math.max(hc.cy, y[i]);
		}
		hc.cx += 1;
		hc.cy += 1;

		/** make graph with vertices **/
		hc.makeGraph(x, y, E);

		System.out.println("\nMatches : " + hc.HopcroftKarp());
	}
}

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 (Integer iterator : collection) {
			if (first)
				first = false;
			else
				writer.print(' ');
			writer.print(iterator);
		}
	}

	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 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 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);
	}
}

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 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;
		}
	}

}

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();
}

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];
		}
		Collections.reverse(path);
		return path;
	}

	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;
		}
		Collections.reverse(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;
	}
}

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

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

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

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

class DFSOrder {
	public final int[] position;
	public final int[] end;

	public DFSOrder(Graph graph) {
		this(graph, 0);
	}

	public DFSOrder(Graph graph, int root) {
		int count = graph.vertexCount();
		position = new int[count];
		end = new int[count];
		int[] edge = new int[count];
		int[] stack = new int[count];
		for (int i = 0; i < count; i++)
			edge[i] = graph.firstOutbound(i);
		stack[0] = root;
		int size = 1;
		position[root] = 0;
		int index = 0;
		while (size > 0) {
			int current = stack[size - 1];
			if (edge[current] == -1) {
				end[current] = index;
				size--;
			} else {
				int next = graph.destination(edge[current]);
				edge[current] = graph.nextOutbound(edge[current]);
				position[next] = ++index;
				stack[size++] = next;
			}
		}
	}

	public DFSOrder(BidirectionalGraph graph) {
		this(graph, 0);
	}

	public DFSOrder(BidirectionalGraph graph, int root) {
		int count = graph.vertexCount();
		position = new int[count];
		end = new int[count];
		int[] edge = new int[count];
		int[] stack = new int[count];
		int[] last = new int[count];
		for (int i = 0; i < count; i++)
			edge[i] = graph.firstOutbound(i);
		stack[0] = root;
		last[root] = -1;
		int size = 1;
		position[root] = 0;
		int index = 0;
		while (size > 0) {
			int current = stack[size - 1];
			if (edge[current] == -1) {
				end[current] = index;
				size--;
			} else {
				int next = graph.destination(edge[current]);
				if (next == last[current]) {
					edge[current] = graph.nextOutbound(edge[current]);
					continue;
				}
				edge[current] = graph.nextOutbound(edge[current]);
				position[next] = ++index;
				last[next] = current;
				stack[size++] = next;
			}
		}
	}
}

abstract class IntervalTree {
	protected int size;

	protected IntervalTree(int size) {
		this(size, true);
	}

	public IntervalTree(int size, boolean shouldInit) {
		this.size = size;
		int nodeCount = Math.max(1, Integer.highestOneBit(size) << 2);
		initData(size, nodeCount);
		if (shouldInit)
			init();
	}

	protected abstract void initData(int size, int nodeCount);

	protected abstract void initAfter(int root, int left, int right, int middle);

	protected abstract void initBefore(int root, int left, int right, int middle);

	protected abstract void initLeaf(int root, int index);

	protected abstract void updatePostProcess(int root, int left, int right,
			int from, int to, long delta, int middle);

	protected abstract void updatePreProcess(int root, int left, int right,
			int from, int to, long delta, int middle);

	protected abstract void updateFull(int root, int left, int right, int from,
			int to, long delta);

	protected abstract long queryPostProcess(int root, int left, int right,
			int from, int to, int middle, long leftResult, long rightResult);

	protected abstract void queryPreProcess(int root, int left, int right,
			int from, int to, int middle);

	protected abstract long queryFull(int root, int left, int right, int from,
			int to);

	protected abstract long emptySegmentResult();

	public void init() {
		if (size == 0)
			return;
		init(0, 0, size - 1);
	}

	private void init(int root, int left, int right) {
		if (left == right) {
			initLeaf(root, left);
		} else {
			int middle = (left + right) >> 1;
			initBefore(root, left, right, middle);
			init(2 * root + 1, left, middle);
			init(2 * root + 2, middle + 1, right);
			initAfter(root, left, right, middle);
		}
	}

	public void update(int from, int to, long delta) {
		update(0, 0, size - 1, from, to, delta);
	}

	protected void update(int root, int left, int right, int from, int to,
			long delta) {
		if (left > to || right < from)
			return;
		if (left >= from && right <= to) {
			updateFull(root, left, right, from, to, delta);
			return;
		}
		int middle = (left + right) >> 1;
		updatePreProcess(root, left, right, from, to, delta, middle);
		update(2 * root + 1, left, middle, from, to, delta);
		update(2 * root + 2, middle + 1, right, from, to, delta);
		updatePostProcess(root, left, right, from, to, delta, middle);
	}

	public long query(int from, int to) {
		return query(0, 0, size - 1, from, to);
	}

	protected long query(int root, int left, int right, int from, int to) {
		if (left > to || right < from)
			return emptySegmentResult();
		if (left >= from && right <= to)
			return queryFull(root, left, right, from, to);
		int middle = (left + right) >> 1;
		queryPreProcess(root, left, right, from, to, middle);
		long leftResult = query(2 * root + 1, left, middle, from, to);
		long rightResult = query(2 * root + 2, middle + 1, right, from, to);
		return queryPostProcess(root, left, right, from, to, middle,
				leftResult, rightResult);
	}
}

class LCA {
	private final long[] order;
	private final int[] position;
	private final IntervalTree lcaTree;
	private final int[] level;

	public LCA(Graph graph) {
		this(graph, 0);
	}

	public LCA(Graph graph, int root) {
		order = new long[2 * graph.vertexCount() - 1];
		position = new int[graph.vertexCount()];
		level = new int[graph.vertexCount()];
		int[] index = new int[graph.vertexCount()];
		for (int i = 0; i < index.length; i++)
			index[i] = graph.firstOutbound(i);
		int[] last = new int[graph.vertexCount()];
		int[] stack = new int[graph.vertexCount()];
		stack[0] = root;
		int size = 1;
		int j = 0;
		last[root] = -1;
		Arrays.fill(position, -1);
		while (size > 0) {
			int vertex = stack[--size];
			if (position[vertex] == -1)
				position[vertex] = j;
			order[j++] = vertex;
			if (last[vertex] != -1)
				level[vertex] = level[last[vertex]] + 1;
			while (index[vertex] != -1
					&& last[vertex] == graph.destination(index[vertex]))
				index[vertex] = graph.nextOutbound(index[vertex]);
			if (index[vertex] != -1) {
				stack[size++] = vertex;
				stack[size++] = graph.destination(index[vertex]);
				last[graph.destination(index[vertex])] = vertex;
				index[vertex] = graph.nextOutbound(index[vertex]);
			}
		}
		lcaTree = new ReadOnlyIntervalTree(order) {
			@Override
			protected long joinValue(long left, long right) {
				if (left == -1)
					return right;
				if (right == -1)
					return left;
				if (level[((int) left)] < level[((int) right)])
					return left;
				return right;
			}

			@Override
			protected long neutralValue() {
				return -1;
			}
		};
		lcaTree.init();
	}

	public int getPosition(int vertex) {
		return position[vertex];
	}

	public int getLCA(int first, int second) {
		return (int) lcaTree.query(Math.min(position[first], position[second]),
				Math.max(position[first], position[second]));
	}

	public int getLevel(int vertex) {
		return level[vertex];
	}

	public int getPathLength(int first, int second) {
		return level[first] + level[second] - 2 * level[getLCA(first, second)];
	}
}

abstract class LongIntervalTree extends IntervalTree {
	protected long[] value;
	protected long[] delta;

	protected LongIntervalTree(int size) {
		this(size, true);
	}

	public LongIntervalTree(int size, boolean shouldInit) {
		super(size, shouldInit);
	}

	@Override
	protected void initData(int size, int nodeCount) {
		value = new long[nodeCount];
		delta = new long[nodeCount];
	}

	protected abstract long joinValue(long left, long right);

	protected abstract long joinDelta(long was, long delta);

	protected abstract long accumulate(long value, long delta, int length);

	protected abstract long neutralValue();

	protected abstract long neutralDelta();

	protected long initValue(int index) {
		return neutralValue();
	}

	@Override
	protected void initAfter(int root, int left, int right, int middle) {
		value[root] = joinValue(value[2 * root + 1], value[2 * root + 2]);
		delta[root] = neutralDelta();
	}

	@Override
	protected void initBefore(int root, int left, int right, int middle) {
	}

	@Override
	protected void initLeaf(int root, int index) {
		value[root] = initValue(index);
		delta[root] = neutralDelta();
	}

	@Override
	protected void updatePostProcess(int root, int left, int right, int from,
			int to, long delta, int middle) {
		value[root] = joinValue(value[2 * root + 1], value[2 * root + 2]);
	}

	@Override
	protected void updatePreProcess(int root, int left, int right, int from,
			int to, long delta, int middle) {
		pushDown(root, left, middle, right);
	}

	protected void pushDown(int root, int left, int middle, int right) {
		value[2 * root + 1] = accumulate(value[2 * root + 1], delta[root],
				middle - left + 1);
		value[2 * root + 2] = accumulate(value[2 * root + 2], delta[root],
				right - middle);
		delta[2 * root + 1] = joinDelta(delta[2 * root + 1], delta[root]);
		delta[2 * root + 2] = joinDelta(delta[2 * root + 2], delta[root]);
		delta[root] = neutralDelta();
	}

	@Override
	protected void updateFull(int root, int left, int right, int from, int to,
			long delta) {
		value[root] = accumulate(value[root], delta, right - left + 1);
		this.delta[root] = joinDelta(this.delta[root], delta);
	}

	@Override
	protected long queryPostProcess(int root, int left, int right, int from,
			int to, int middle, long leftResult, long rightResult) {
		return joinValue(leftResult, rightResult);
	}

	@Override
	protected void queryPreProcess(int root, int left, int right, int from,
			int to, int middle) {
		pushDown(root, left, middle, right);
	}

	@Override
	protected long queryFull(int root, int left, int right, int from, int to) {
		return value[root];
	}

	@Override
	protected long emptySegmentResult() {
		return neutralValue();
	}
}

class SumIntervalTree extends LongIntervalTree {
	public SumIntervalTree(int size) {
		super(size);
	}

	@Override
	protected long joinValue(long left, long right) {
		return left + right;
	}

	@Override
	protected long joinDelta(long was, long delta) {
		return was + delta;
	}

	@Override
	protected long accumulate(long value, long delta, int length) {
		return value + delta * length;
	}

	@Override
	protected long neutralValue() {
		return 0;
	}

	@Override
	protected long neutralDelta() {
		return 0;
	}
}

abstract class ReadOnlyIntervalTree extends IntervalTree {
	protected long[] value;
	protected long[] array;

	protected ReadOnlyIntervalTree(long[] array) {
		super(array.length, false);
		this.array = array;
		init();
	}

	@Override
	protected void initData(int size, int nodeCount) {
		value = new long[nodeCount];
	}

	@Override
	protected void initAfter(int root, int left, int right, int middle) {
		value[root] = joinValue(value[2 * root + 1], value[2 * root + 2]);
	}

	@Override
	protected void initBefore(int root, int left, int right, int middle) {
	}

	@Override
	protected void initLeaf(int root, int index) {
		value[root] = array[index];
	}

	@Override
	protected void updatePostProcess(int root, int left, int right, int from,
			int to, long delta, int middle) {
		throw new UnsupportedOperationException();
	}

	@Override
	protected void updatePreProcess(int root, int left, int right, int from,
			int to, long delta, int middle) {
		throw new UnsupportedOperationException();
	}

	@Override
	protected void updateFull(int root, int left, int right, int from, int to,
			long delta) {
		throw new UnsupportedOperationException();
	}

	@Override
	protected long queryPostProcess(int root, int left, int right, int from,
			int to, int middle, long leftResult, long rightResult) {
		return joinValue(leftResult, rightResult);
	}

	@Override
	protected void queryPreProcess(int root, int left, int right, int from,
			int to, int middle) {
	}

	@Override
	protected long queryFull(int root, int left, int right, int from, int to) {
		return value[root];
	}

	@Override
	protected long emptySegmentResult() {
		return neutralValue();
	}

	@Override
	public void update(int from, int to, long delta) {
		throw new UnsupportedOperationException();
	}

	@Override
	protected void update(int root, int left, int right, int from, int to,
			long delta) {
		throw new UnsupportedOperationException();
	}

	protected abstract long neutralValue();

	protected abstract long joinValue(long left, long right);
}