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

N,A,B=map(int,input().split())

if N==1:
    print("YES")
    print(A,B)
    sys.exit()

BIT=0
for i in range(N):
    #print(1<<i,(1<<i) & A,(1<<i) & B)
    if (1<<i) & A!=(1<<i) & B:
        BIT+=1

if BIT%2==1:
    print("YES")
else:
    print("NO")
    sys.exit()


check=[0]*(1<<N)

ANS=[]
check[A]=1
check[B]=1
W=[N]*(1<<N)

for j in range(N):
    W[A^(1<<j)]-=1
    #W[B^(1<<j)]-=1

NOW=A
#print(W,NOW)
for i in range((1<<N)-2):
    ANS.append(NOW)
    
    #WN=N
    NEXTS=[]
    for j in range(N):
        if check[NOW ^ (1<<j)]==0:
            NEXTS.append([W[NOW ^ (1<<j)],NOW ^ (1<<j)])

    MIN=min(NEXTS)[0]
    NEXT2=[n[1] for n in NEXTS if n[0]==MIN]

    BITS=0

    for n in NEXT2:
        BBITS=0
        for j in range(N):           
            if A & (1<<j)==(n & (1<<j)):
                BBITS+=1

        if BBITS>=BITS:
            NEXT=n
            BBITS=BITS
            
    for j in range(N):
        W[NEXT^(1<<j)]-=1

    NOW=NEXT
    check[NOW]=1

    #print(W,NOW)
ANS.append(NOW)
ANS.append(B)

print(*ANS)