Problem: We have a pyramid with N steps, built with blocks. The steps are numbered 1 through N from top to bottom. For each 1≤i≤N, step i consists of 2i-1 blocks aligned horizontally. The pyramid is built so that the blocks at the centers of the steps are aligned vertically.

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A pyramid with N=4 steps

Snuke wrote a permutation of (1, 2, ..., 2N-1) into the blocks of step N. Then, he wrote integers into all remaining blocks, under the following rule:

* The integer written into a block b must be equal to the median of the three integers written into the three blocks directly under b, or to the lower left or lower right of b.



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Writing integers into the blocks

Afterwards, he erased all integers written into the blocks. Now, he only remembers that the integer written into the block of step 1 was x.

Construct a permutation of (1, 2, ..., 2N-1) that could have been written into the blocks of step N, or declare that Snuke's memory is incorrect and such a permutation does not exist.

Constraints

* 2≤N≤10^5
* 1≤x≤2N-1

Input

The input is given from Standard Input in the following format:


N x


Output

If no permutation of (1, 2, ..., 2N-1) could have been written into the blocks of step N, print `No`.

Otherwise, print `Yes` in the first line, then print 2N-1 lines in addition.

The i-th of these 2N-1 lines should contain the i-th element of a possible permutation.

Examples

Input

4 4


Output

Yes
1
6
3
7
4
5
2


Input

2 1


Output

No

Solution (in JAVA):

Solution:
import java.util.HashMap;
import java.util.Map;
import java.util.Scanner;

public class Main {
    public static void main(String args[] ) throws Exception {
    	new Main().exec(args);
    }

    public Main() {
    }

    public void exec(String args[]) throws Exception {
    	Scanner sc = new Scanner(System.in);
    	int N=sc.nextInt();
    	int x=sc.nextInt();
    	boolean flg=true;
    	if(x==1 || x==2*N-1) flg=false;
    	Map<Integer,Integer> output = new HashMap<Integer,Integer>();
    	Map<Integer,Integer> outputVal = new HashMap<Integer,Integer>();
    	System.out.println(flg?"Yes":"No");
    	if(flg) {
    		if(N==2) {
    			output.put(1,1);
    			output.put(2,2);
    			output.put(3,3);
    		}
    		else {
    			if(x==2) {
            		put(output,outputVal, N-1,2*N-1);
            		put(output,outputVal, N,x);
            		put(output,outputVal, N+1,1);
            	}else{
            		put(output,outputVal, N-2,2);
            		put(output,outputVal, N-1,2*N-1);
        			put(output,outputVal, N,x);
            		put(output,outputVal, N+1,1);
            	}
    		}
    		output(output, 2*N-1);
    	}

    }
    private void output(Map<Integer,Integer> map, int max) {
    	StringBuilder sb = new StringBuilder();
		for(int i=1; i<=max; i++) {
			if(map.containsKey(i)){
				sb.append(map.get(i)).append("\n");
			}else{
				sb.append(i).append("\n");
			}
		}
		System.out.println(sb);
    }
    private void put(Map<Integer,Integer>  map,Map<Integer,Integer>  valmap, int index, int val) {
    	int tmp_idxs_val = index;
    	int tmp_vals_idx = val;
    	if(map.containsKey(index)){
    		tmp_idxs_val = map.get(index);
    	}
    	if(valmap.containsKey(val)){
    		tmp_vals_idx = valmap.get(val);
    	}
    	map.put(index, val);
    	valmap.put(val, index);
    	map.put(tmp_vals_idx, tmp_idxs_val);
    	valmap.put(tmp_idxs_val, tmp_vals_idx);
    }
}