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.

<image>

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.



<image>

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 CPP):

Solution:
#include<bits/stdc++.h>
using namespace std;
const int N=1e6+5;
int a[N],vis[N];
int main()
{
    int n,x;scanf("%d%d",&n,&x);
    if(x==1||x==2*n-1){printf("No\n");return 0;}
    printf("Yes\n");
    if(n==2){printf("1\n2\n3\n");return 0;}
    if(x==2*n-2)a[n-1]=x-2,a[n]=x,a[n+1]=x+1,a[n+2]=x-1,vis[x-2]=1;
    else a[n-1]=x+2,a[n]=x,a[n+1]=x-1,a[n+2]=x+1,vis[x+2]=1;
    vis[x]=vis[x+1]=vis[x-1]=1;
    int k=1;
    for(int i=1;i<=2*n-1;i++)
    {
        if(a[i])continue;
        while(vis[k])k++;a[i]=k;vis[k]=1;
    }
    for(int i=1;i<=2*n-1;i++)printf("%d\n",a[i]);
}