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.

<image>

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.



<image>

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 PYTHON):

Solution:
import sys
from collections import deque
import copy

from math import *

def get_read_func(fileobject):
    if fileobject == None :
        return raw_input
    else:
        return fileobject.readline


def main():
    if len(sys.argv) > 1:
        f = open(sys.argv[1])
    else:
        f = None
    read_func = get_read_func(f);
    input_raw = read_func().strip().split()
    [N, x] =  [long(input_raw[0]), long(input_raw[1])]
    if x <= 1 or x >= 2*N - 1:
        print 'No'

    else:
        print 'Yes'
        l = [0 for i in range(2*N - 1)]
        m = (2*N-1)/2
        l[m] = x
        already_set = set([x])


        if x + 2 <= 2 * N - 1:
            if m - 1 >= 0:
                l[m - 1] = x + 1
                already_set.add(x + 1)
            if m + 1 < 2*N - 1:
                l[m + 1] = x - 1
                already_set.add(x - 1)
            if m + 2 < 2*N - 1:
                l[m + 2] = x + 2
                already_set.add(x + 2)
        else:
            if m - 1 >= 0:
                l[m - 1] = x - 1
                already_set.add(x - 1)
            if m + 1 < 2*N - 1:
                l[m + 1] = x + 1
                already_set.add(x + 1)
            if m + 2 < 2*N - 1:
                l[m + 2] = x - 2
                already_set.add(x - 2)

        a = 1
        for i in range(2 * N - 1):
            if l[i] == 0:
                while a <= 2*N - 1 :
                    if a not in already_set:
                        print a
                        a += 1
                        break;
                    else:
                        a += 1
            else:
                print l[i]



if __name__ == '__main__':
    main()