atcoder#AGC006B. [AGC006B] Median Pyramid Easy

[AGC006B] Median Pyramid Easy

Score : 400400 points

Problem Statement

We have a pyramid with NN steps, built with blocks. The steps are numbered 11 through NN from top to bottom. For each 1iN1 \leq i \leq N, step ii consists of 2i12i-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=4N=4 steps

Snuke wrote a permutation of (11, 22, ......, 2N12N-1) into the blocks of step NN. Then, he wrote integers into all remaining blocks, under the following rule:

  • The integer written into a block bb must be equal to the median of the three integers written into the three blocks directly under bb, or to the lower left or lower right of bb.

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 11 was xx.

Construct a permutation of (11, 22, ......, 2N12N-1) that could have been written into the blocks of step NN, or declare that Snuke's memory is incorrect and such a permutation does not exist.

Constraints

  • 2N1052 \leq N \leq 10^5
  • 1x2N11 \leq x \leq 2N-1

Input

The input is given from Standard Input in the following format:

NN xx

Output

If no permutation of (11, 22, ......, 2N12N-1) could have been written into the blocks of step NN, print No.

Otherwise, print Yes in the first line, then print 2N12N-1 lines in addition.

The ii-th of these 2N12N-1 lines should contain the ii-th element of a possible permutation.

4 4
Yes
1
6
3
7
4
5
2

This case corresponds to the figure in the problem statement.

2 1
No

No matter what permutation was written into the blocks of step NN, the integer written into the block of step 11 would be 22.