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
input = sys.stdin.buffer.readline
N, x = map(int, input().split())
if N == 2:
  if x == 2:
    print("Yes")
    print(1)
    print(2)
    print(3)
  else:
    print("No")
  exit(0)
if x == 1 or x == 2 * N - 1:
  print("No")
  exit(0)
print("Yes")
if x == 2:
  i = 4
  for _ in range(1, N - 2):
    i += 1
    print(i)
  print(x + 2)
  print(x - 1)
  print(x)
  print(x + 1)
  for _ in range(N + 2, 2 * N):
    i += 1
    print(i)
else:
  i = 0
  for _ in range(1, N - 1):
    if i == x - 3:
      i += 5
    else:
      i += 1
    print(i)
  print(x - 1)
  print(x)
  print(x + 1)
  print(x - 2)
  for _ in range(N + 3, 2 * N):
    if i == x - 3:
      i += 5
    else:
      i += 1
    print(i)