Problem Statement

You are given a positive integer N and $a sequence A = (A_0, A_1, \ldots, A_{2^N-1})$ of 2^N terms, where each A_i is an integer between 0 and 2^N-1 (inclusive) and $A_i\neq A_j$ holds if $i\neq j$.
You can perform the following two kinds of operations on A:

• Operation +: For every i, change A_i to $(A_i + 1) \bmod 2^N$.
• Operation $\oplus$: Choose an integer x between 0 and 2^N-1. For every i, change A_i to $A_i\oplus x$.

Here, $\oplus denotes bitwise \mathrm{XOR}$.

    What is bitwise $\mathrm{XOR}$?


    The bitwise $\mathrm{XOR}$ of non-negative integers A and $B, A \oplus B$, is defined as follows:

• When $A \oplus B$ is written in base two, the digit in the 2^k'$s place (k \geq 0)$ is 1 if exactly one of A and B is 1, and 0 otherwise.

  For example, we have $3 \oplus 5 = 6 ($in base two: $011 \oplus 101 = 110)$.
  

Your objective is to make A_i = i for every i. Determine whether it is achievable. It can be proved that, under the Constraints of this problem, one can achieve the objective with at most 10^6 operations if it is achievable at all. Print such a sequence of operations.

Input

Input is given from Standard Input in the following format:
N
$A_0 A_1 \ldots A_{2^N - 1}$

Output

If it is possible to make A_i = i for every i, print Yes; otherwise, print No.
In the case of Yes, it should be followed by a sequence of operations that achieves the objective, in the following format:
K
$x_1 x_2 \ldots x_K$

Here, K is a non-negative integer representing the number of operations, which must satisfy $0\leq K\leq 10^6; x_i$ is an integer representing the i-th operation, which should be set as follows.

• If the i-th operation is Operation +, x_i=-1.
• If the i-th operation is Operation $\oplus, x_i$ should be the integer chosen in that operation.

If there are multiple possible solutions, you may print any of them.

Constraints

• $1\leq N\leq 18$
• $0\leq A_i \leq 2^N - 1$
• $A_i\neq A_j$, if $i\neq j$

Sample Input 1

3
5 0 3 6 1 4 7 2

Sample Output 1

Yes
4
-1 6 -1 1

The sequence of operations above changes the sequence A as follows.

• Initially: A = (5,0,3,6,1,4,7,2)
• Operation +: A = (6,1,4,7,2,5,0,3)
• Operation $\oplus (x = 6)$：A = (0,7,2,1,4,3,6,5)
• Operation +: A = (1,0,3,2,5,4,7,6)
• Operation $\oplus (x = 1)$：A = (0,1,2,3,4,5,6,7)

Sample Input 2

3
2 5 4 3 6 1 0 7

Sample Output 2

No

No sequence of operations achieves the objective.

Sample Input 3

3
0 1 2 3 4 5 6 7

Sample Output 3

Yes
0

Performing zero operations achieves the objective. Whether an empty line is printed or not does not affect the verdict.

见题面 Input 部分

见题面 Output 部分