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 PyPy3 (2.4.0) (judge verdict: Accepted)

n, a, b = [int(i) for i in input().split(" ")]

def p2l(x):
    l = []
    for i in range(0, n):
        l = [x % 2] + l
        x = x // 2
    return l

def l2p(x):
    return sum([x[i] * 2**(len(x) - i - 1) for i in range(0, len(x))])

al = p2l(a)
bl = p2l(b)

def gray(a, b):
    if len(a) == 1:
        return [a, b]
    
    for i in range(0, len(a)):
        if a[i] != b[i]:
            break
    ah = a[i]
    bh = b[i]
    del a[i]
    del b[i]
    
    c = [1 - a[0]] + a[1:]
    g1 = gray(a[:], c[:])
    g2 = gray(c[:], b[:])
    g1 = [h[0:i] + [ah] + h[i:] for h in g1]
    g2 = [h[0:i] + [bh] + h[i:] for h in g2]
    return g1 + g2

if sum(al) % 2 != sum(bl) % 2:
    print("YES")
    g = gray(al, bl)
    p = [l2p(gi) for gi in g]
    print(' '.join(map(str, p)))
else:
    print("NO")