Problem:
Score : 800 points

Problem Statement

You are given integers N,\ A and B.
Determine if there exists a permutation (P_0,\ P_1,\ ...\ P_{2^N-1}) of (0,\ 1,\ ...\ 2^N-1) that satisfies all of the following conditions, and create one such permutation if it exists.

P_0=A

P_{2^N-1}=B

For all 0 \leq i < 2^N-1, the binary representations of P_i and P_{i+1} differ by exactly one bit.

Constraints

1 \leq N \leq 17

0 \leq A \leq 2^N-1

0 \leq B \leq 2^N-1

A \neq B

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N A B

Output

If there is no permutation that satisfies the conditions, print NO.

If there is such a permutation, print YES in the first line.
Then, print (P_0,\ P_1,\ ...\ P_{2^N-1}) in the second line, with spaces in between.
If there are multiple solutions, any of them is accepted.

Sample Input 1

2 1 3

Sample Output 1

YES
1 0 2 3

The binary representation of P=(1,0,2,3) is (01,00,10,11), where any two adjacent elements differ by exactly one bit.

Sample Input 2

3 2 1

Sample Output 2

NO

Solution in Ruby (2.3.3) (judge verdict: Wrong Answer)

n, a, b = gets.split.map(&:to_i)

bitdiff = (a ^ b).to_s(2)

def func(allbits, diffbits)
  return allbits if allbits.size == 1
  p allbits, diffbits
  if diffbits.size == 1
    lr = (allbits - diffbits).inject([]) do |memo, item|
      memo + [item] + memo
    end
    lr + diffbits + lr
  else
    left, right, center = diffbits.each_slice(diffbits.size/2).map(&:itself)
    newallbits = allbits - center
    func(newallbits, left) + center + func(newallbits, right)
  end
end

unless bitdiff.count('1').to_s(2).include?('0')
  puts 'YES'
  ans = []
  diffbits = bitdiff.reverse.each_char.each_with_index.select {|item, index| item == '1'}.map(&:last)
  ans = func((0..(n-1)).to_a, diffbits).inject([a]) do |memo, item|
    memo << (memo.last ^ 2**(item))
  end
  puts ans.join(' ')
else
  puts 'NO'
end