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 n,x,i,j,r[200200];
bool u[200200];
void st(int x, int v) {
  r[x]=v; u[v]=true;
}
int main() {
  scanf("%d%d",&n,&x);
  if (x==1 || x==2*n-1) { puts("No"); return 0; }
  st(n,x);
  if (x==2) {
    st(n-1,1);
    st(n+1,3);
    if (n>2) st(n-2,4);
  } else {
    st(n-1,x+1);
    st(n+1,x-1);
    if (n>2) st(n-2,x-2);
  }
  puts("Yes");
  for (i=1; i<2*n; i++) if (r[i]==0) {
    for (j++; u[j]; j++);
    printf("%d\n",j);
  } else printf("%d\n",r[i]);
  return 0;
}