Problem: Gerald has a friend, Pollard. Pollard is interested in lucky tickets (ticket is a sequence of digits). At first he thought that a ticket is lucky if between some its digits we can add arithmetic signs and brackets so that the result obtained by the arithmetic expression was number 100. But he quickly analyzed all such tickets and moved on to a more general question. Now he explores k-lucky tickets.

Pollard sais that a ticket is k-lucky if we can add arithmetic operation signs between its digits to the left or right of them (i.e., "+", "-", " × ") and brackets so as to obtain the correct arithmetic expression whose value would equal k. For example, ticket "224201016" is 1000-lucky as ( - 2 - (2 + 4)) × (2 + 0) + 1016 = 1000.

Pollard was so carried away by the lucky tickets that he signed up for a seminar on lucky tickets and, as far as Gerald knows, Pollard will attend it daily at 7 pm in some famous institute and will commute to it in the same tram for m days. In this tram tickets have eight digits. And Gerald wants to make a surprise for Pollard: each day Pollard will receive a tram k-lucky ticket. The conductor has already agreed to give Pollard certain tickets during all these m days and he only wants Gerald to tell him what kind of tickets to give out. In this regard, help Gerald pick exactly m distinct k-lucky tickets.

Input

The single line contains two integers k and m (0 ≤ k ≤ 104, 1 ≤ m ≤ 3·105).

Output

Print m lines. Each line must contain exactly 8 digits — the k-winning ticket. The tickets may begin with 0, all tickets must be distinct. If there are more than m distinct k-lucky tickets, print any m of them. It is guaranteed that at least m distinct k-lucky tickets exist. The tickets can be printed in any order.

Examples

Input

0 3


Output

00000000
00000001
00000002


Input

7 4


Output

00000007
00000016
00000017
00000018

Solution (in JAVA):

Solution:
import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.HashMap;
import java.util.HashSet;
import java.util.InputMismatchException;
import java.util.List;
import java.util.Map;
import java.util.Set;

public class E {
	static Set<String> ans;
	static Map<Key, Set<Integer>> permed;
	static Set<Key> finalPermed;
	
	static int m;
	public static void main(String[] args) throws Exception{
		InputReader in = new InputReader(System.in);
		PrintWriter out = new PrintWriter(System.out);
		
		int k = in.nextInt();
		m = in.nextInt();
		permed = new HashMap<Key, Set<Integer>>();
		finalPermed = new HashSet<Key>();
		
		String[] arr = new String[]{"","0","00","000","0000","00000","000000","0000000","00000000"};
		
		ans = new HashSet<String>();
		
		for(int r = 1234; r <= 9999; r++){
			//get all numbers that can be formed by playing around with r's digits
			String rStr = "" + r;
			rStr += arr[4-rStr.length()];
			int[] digs = new int[4];
			for(int i =0; i < digs.length;i++)
				digs[i] = rStr.charAt(i)-'0';
			//permute, stick, operation
			Set<Integer> vals = new HashSet<Integer>();
			stick(0, digs, new ArrayList<Integer>(), vals);
			//for each such number n, do k-n. Since r -> n, then k-n and n = k
			for(int n : vals){
				int other = Math.abs(k-n);
				String otherStr = "" + other;
				if(otherStr.length() + rStr.length() > 8) continue;
				String rem = arr[8-otherStr.length()-rStr.length()];
				ans.add(otherStr + rStr + rem);
				ans.add(otherStr + rem + rStr);
				ans.add(rStr + otherStr + rem);
				ans.add(rStr + rem + otherStr);
				if(ans.size() > m)break;
			}
			if(ans.size() > m)break;
		}
		//System.out.println(ans.size());
		int i = 0;
		for(String s : ans){
			out.println(s);
			i++;
			if(i == m)break;
		}
		out.close();
	}
	
	static void stick(int toSet, int[] ordered, List<Integer> sticked, Set<Integer> vals){
		if(toSet == ordered.length){
			operation(0, 0, sticked, vals);
		}else{
			//stick to prev
			if(toSet != 0){
				int prevVal = sticked.get(sticked.size()-1);
				int newSticked = prevVal*10 + ordered[toSet];
				sticked.set(sticked.size()-1, newSticked);
				stick(toSet+1, ordered, sticked, vals);
				sticked.set(sticked.size()-1, prevVal);
			}
			
			//don't stick to prev
			sticked.add(ordered[toSet]);
			stick(toSet+1, ordered, sticked, vals);
			sticked.remove(sticked.size()-1);
		}
	}
	
	static void operation(int toSet, int curVal, List<Integer> sticked, Set<Integer> vals){
		if(toSet == sticked.size()){
			vals.add(Math.abs(curVal));
		}else{
			if(toSet == 0){
				operation(toSet+1, sticked.get(toSet), sticked, vals);
			}else{
				//sub
				operation(toSet+1, curVal - sticked.get(toSet), sticked, vals);
				//add
				operation(toSet+1, curVal + sticked.get(toSet), sticked, vals);
				//mul
				operation(toSet+1, curVal * sticked.get(toSet), sticked, vals);				
			}
		}
	}
	
	static class Key{
		int[] arr;

		public Key(int[] arr) {
			this.arr = arr.clone();
		}

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

		@Override
		public boolean equals(Object obj) {
			Key other = (Key) obj;
			return Arrays.equals(arr, other.arr);
		}
	}
	
	static 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 nextInt() {
			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 nextLong() {
			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 nextToken() {
			int c = read();
			while (isSpaceChar(c))
				c = read();
			StringBuilder res = new StringBuilder();
			do {
				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;
		}

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