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 CPP):

Solution:
#include<bits/stdc++.h>
using namespace std;
int main(){
  long N,x;cin>>N>>x;
  if(x==1||x==2*N-1){cout<<"No";return 0;}
  cout<<"Yes"<<endl;
  
  if(N==2){cout<<1<<endl<<2<<endl<<3;return 0;}

  vector<long>A(2*N-1,-1);
  long s=(x<=N)*2-1;
  A[N-3]=x+2*s; //x+2<=2*N-1
  A[N-2]=x-1*s; // 
  A[N-1]=x;
  A[N]=x+1*s; 
  
  long p=0;
  for(long i=1;i<=2*N-1;i++){
    while(p<2*N-1&&A[p]!=-1)p++;
    if(i==x-1*s||i==x||i==x+1*s||i==x+2*s)continue;
    A[p]=i;
  } 
  for(long x:A)cout<<x<<endl;
}