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 Haskell (GHC 7.10.3) (judge verdict: Wrong Answer)

import Data.Bits
import qualified Data.Vector.Unboxed as V
import Data.List
import Data.Monoid

path :: Int -> [Int] -> V.Vector Int
path 1 [k] = V.fromList [0,2^k]
path n (k:ks) = let v = path (n - 1) ks
                    (v0,v1) = V.splitAt (2^(n-2)) v
                    y = 2^k
                in v0 <> V.map (`xor` y) (V.reverse v0) <> V.map (`xor` y) (V.reverse v1) <> v1

parseInts :: String -> [Int]
parseInts s = case reads s of
                [(x,t)] -> x : parseInts t
                _ -> []

myAssert :: Bool -> IO ()
myAssert True = return ()
myAssert False = do
  let v = V.replicate (1024 * 1024 * 1024) (42 :: Int)
  print (V.length v)

main = do
  [n,a,b] <- parseInts <$> getLine
  let c = a `xor` b
  if popCount c == 1
    then do putStrLn "YES"
            let k = countTrailingZeros c
            myAssert (c == 2^k)
            let xs = V.map (xor a) $ path n (filter (/= k) [0..n-1] ++ [k])
                xss = V.toList xs
            putStrLn $ intercalate " " $ map show xss
            let differByOne a b = popCount (a `xor` b) == 1
            myAssert (and $ zipWith differByOne xss (tail xss))
            myAssert (a == V.head xs && b == V.last xs)
            myAssert (sort xss == [0..2^n-1])
    else putStrLn "NO"