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)
dif = Array.new(n,0)
for i in 0..n-1
  if a[i] < b[i]
    dif[i] = 1
  elsif a[i] > b[i]
    dif[i] = -1
  end
end
if ((dif.count(1)+dif.count(-1)) % 2) == 0 || dif.count(1)+dif.count(-1) > 2**n-1
  puts "NO"
  exit
end
check = Array.new(2**n-1,0)
ans = Array.new(2**n,-1)
ans[0] = a
check[a] = 1
check[b] = 1
verse = 1
for i in 1..2**n-2
  if ans[i-1] != 0 && verse == 1
    for j in 0..n-1
      if ans[i-1][j] == 1
        ans[i] = ans[i-1] - (2**j)
        check[ans[i]] = 1
        verse -= 1
        break
      end
    end
  else
    verse = 0
    for j in 0..n-1
      if ans[i-1][j] == 1
        if check[ans[i-1]-(2**j)] == 0
          ans[i] = ans[i-1] - (2**j)
          check[ans[i]] = 1
          break
        end
      else
        if check[ans[i-1]+(2**j)] == 0
          ans[i] = ans[i-1] + (2**j)
          check[ans[i]] = 1
          break
        end
      end
    end
  end
  if ans[i] == -1
    puts "NO"
    exit
  end
end
ans[-1] = b
puts "YES"
for i in 0..2**n - 2
  print "#{ans[i]} "
end
puts b