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 Perl (v5.18.2) (judge verdict: Wrong Answer)

#!/usr/bin/perl
my ($n,$a,$b) = map { $_ - 0 } split(/\s+/,<STDIN>);
my $n2 = 2 ** $n;
my %fac = ();
for(my $i=0;$i<18;$i++){
  my $ii = 2 ** $i;
  $fac{$ii} = 1;
}
my $xab = ( $a ^ $b );
unless( $fac{$xab} ){
  print "NO\n"; exit(0);
}
my @bl = &mkl($n);
my $str_a = sprintf("%0${n}b",$a);
my $str_b = sprintf("%0${n}b",$b);
my $ii = -1;
for(my $i=0;$i<=$#bl;$i++){
  my $j = ( $n2 + $i - 1 ) % $n2;
  if( ( oct('0b' . $bl[$i]) ^ oct('0b' . $bl[$j]) ) == $xab ){
    $ii = $i;
    last;
  }
}
my $orig_s = $bl[$ii];
#print "orig=$orig_s\n";
if( $ii < 0 ){
  print "NO\n"; exit(0);
}
print "YES\n";
my @l=();
$#l = $#bl;
for(my $i=0;$i<=$#bl;$i++){
  my $s = $bl[ ($n2 + $ii + $i )%$n2 ];
  my $ss = '';
  for(my $j=0;$j<$n;$j++){
    if( substr($orig_s,$j,1) ne substr($str_a,$j,1) ){
      $ss .= ( substr($s,$j,1) eq '0' ? '1' : '0' );
    } else {
      $ss .= substr($s,$j,1);
    }
  }
  #print "s=$s,ss=$ss\n";
  $l[$i] = oct("0b$ss");
}
#print "l=" . join(":",@l) . "\n";

for(my $i=0;$i<$n2;$i++){
  print " " if $i>0;
  print $l[$i];
}
print "\n";
exit(0);

sub mkl {
  my $lv = shift;
  if( $lv == 1 ){
    return ('0','1');
  }
  my @a = (&mkl($lv-1));
  my @ra = (reverse(@a));
  my @r = ( (map { '0' . $_ } @a) , (map { "1$_" } @ra) );
  return @r;
}