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Contest: Task: Related: TaskA TaskC

Score : $400$ points

Problem Statement

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.

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 $b$.

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.

Sample Input 1

4 4

Sample Output 1

Yes
1
6
3
7
4
5
2

This case corresponds to the figure in the problem statement.

Sample Input 2

2 1

Sample Output 2

No

No matter what permutation was written into the blocks of step $N$, the integer written into the block of step $1$ would be $2$.