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;
typedef long long ll;
ll a[200005];

int main(){
  ll n,x;
  cin>>n>>x;
  if(x==n*2-1 || x==1) cout<<"No"<<endl;
  else{
    cout<<"Yes"<<endl;
    if(x-2<1) {a[3]=x+2;a[0]=x;a[1]=x-1;a[2]=x*2-1;}
    else {a[0]=x-2;a[1]=x;a[2]=x-1;a[3]=x*2-1;}
    
    
    for(int i=1;i<=(n*2-1);i++) a[i] = i;
    swap(a[n],a[x]);
    swap(a[n-1],a[1]);
    swap(a[n+1],a[n*2-1]);
    if(x-2>1) swap(a[n+2],a[x-2]);
    else swap(a[n-2],a[x+2]);
    for(int i=1;i<=(n*2-1);i++) cout<<a[i]<<endl;
  }
  return 0;
}