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 Fortran (gfortran v4.8.4) (judge verdict: Wrong Answer)

program differ_by_1_bit
  implicit none
  integer :: n, a, b, i, m, p(0:131071) = 0
  read(*,*) n, a, b
  if (mod(popcnt(a),2) == mod(popcnt(b),2)) then
    write(*,'(a)') "NO"
    stop
  end if
  m = lshift(1,n)-1
  p(0:m) = compose(n,a,b)
  write(*,'(a)') "YES"
  write(*,'(i0)',advance='no') p(0)
  do i = 1, m
    write(*,'(x,i0)',advance='no') p(i)
  end do
  write(*,*)
  stop
contains
  recursive function compose(n,a,b) result(p)
    implicit none
    integer, intent(in) :: n, a, b
    integer :: s, t, x = 0, y = 0, c = 0, i, j, k
    integer :: p(0:lshift(1,n)-1), q(0:lshift(1,n-1)-1), r(0:lshift(1,n-1)-1)
    p = 0
    if (n == 1) then
      p(0:1) = (/a,b/)
      return
    end if
    s = lshift(1,n-1)-1
    t = lshift(1,n)-1
    do i = 0, n-1
      if (btest(a,i) .eqv. btest(b,i)) cycle
      do j = 0, i-1
        if (btest(a,j)) x = ibset(x,j)
        if (btest(b,j)) y = ibset(y,j)
      end do
      k = i
      do j = i+1, n-1
        if (btest(a,j)) x = ibset(x,j-1)
        if (btest(b,j)) y = ibset(y,j-1)
      end do
      c = ibset(x,0)
      if (btest(x,0)) c = ibclr(x,0)
      exit
    end do
    q = compose(n-1,x,c)
    r = compose(n-1,c,y)
    do i = 0, s
      do j = 0, k-1
        if (btest(q(i),j)) p(i) = ibset(p(i),j)
        if (btest(r(i),j)) p(i+s+1) = ibset(p(i+s+1),j)
      end do
      if (btest(a,k)) p(i) = ibset(p(i),k)
      if (btest(b,k)) p(i+s+1) = ibset(p(i+s+1),k)
      do j = k+1, n-1
        if (btest(q(i),j-1)) p(i) = ibset(p(i),j)
        if (btest(r(i),j-1)) p(i+s+1) = ibset(p(i+s+1),j)
      end do
    end do
    return
  end function compose
end program differ_by_1_bit