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 PYTHON3):

Solution:

import sys
if __name__ == "__main__":

	N,X = map(int, input().split())
	res = 2 * N - 1
	if (X == 1 or X == 2 * N - 1):
		print ("No")
		sys.exit()
	result = [0] * res
	left = X - 1
	result[N-1] = X
	right = X + 1
	left_idx = N - 2
	right_idx = N

	cnt = 0
	while (left > 0 and right < 2 * N):
		if (cnt % 2 == 0):
			result[left_idx] = left
			result[right_idx] = right
		else:
			result[left_idx] = right
			result[right_idx] = left

		left_idx -= 1
		left -= 1

		right_idx += 1
		right += 1

		cnt += 1
	if (left > 0):
		for x in range(left_idx, -1, -1):
			result[x] = left
			left -= 1
		for x in range(right_idx, 2 * N-1):
			result[x] = left
			left -= 1
	if (right < 2 * N):
		for x in range(left_idx, -1, -1):
			result[x] = right
			right += 1
		for x in range(right_idx, 2 * N-1):
			result[x] = right
			right += 1

	print ("Yes")
	for x in result:
		print (x)