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: Wrong Answer)

import sys
sys.setrecursionlimit(20000000)
from copy import deepcopy
input = sys.stdin.readline
n,a,b = map(int,input().split())
if max(a,b)-2**(n-1) != min(a,b):
    print("NO")
    exit()
syo = min(a,b)
dai = max(a,b)
ans = [0,1]
hi = True
ima = 1
while syo:
    if syo%2 == 1:
        tmp = deepcopy(ans)
        for i in range(len(tmp)):
            tmp[i] += 2**ima
        if hi:
            ans = tmp[::-1]+ans
            hi = False
        else:
            hi=True
            ans = ans + tmp[::-1]
    else:
        tmp = deepcopy(ans)
        for i in range(len(tmp)):
            tmp[i] += 2**ima
        if hi:
            ans = ans+tmp[::-1]
        else:
            ans = tmp[::-1]+ans
        
    syo//=2
    ima+=1
if ans[0] != min(a,b):
    ans = ans[::-1]
while ans[-1] != dai:
    tmp = deepcopy(ans)
    for i in range(len(tmp)):
        tmp[i] += 2**ima
    ans += tmp[::-1]
    ima+=1
print("YES")
if ans[0] != a:
    ans = ans[::-1]
for i in range(2**n-1):
    if abs(bin(ans[i+1]).count("1") -   bin(ans[i]).count("1") ) != 1:
        print(ans[i],ans[i+1])
        assert oi
print(*ans)