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: Accepted)

n,a,b = gets.split.map(&:to_i)

def rec(n,a,b)
    return [a,b] if n == 1
    x = 0
    n.times do |i|
        if (a[i]^b[i]) == 1
            x = i
            break
        end
    end
    p = 0
    x.times do |i|
        p |= 1<<i
    end
    q = 0
    for i in x+1..n-1
        q |= 1<<i
    end
    c = ((a&p)|((a&q)>>1))
    d = ((b&p)|((b&q)>>1))
    u = rec(n-1,c,c^1)
    v = rec(n-1,c^1,d)
    q >>= 1
    y = a[x]
    (1<<(n-1)).times do |i|
        u[i] = (u[i]&p)|((u[i]&q)<<1)|(y<<x)
        v[i] = (v[i]&p)|((v[i]&q)<<1)|((y^1)<<x)
    end
    return u+v
end
x = 0
n.times do |i|
    x ^= a[i]^b[i]
end
if x == 0
    puts "NO"
else
    puts "YES"
    puts rec(n,a,b).join(" ")
end