Score : $1100$ points

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.

3 \oplus 5 = 6$$$$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.

Constraints

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

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.

3
5 0 3 6 1 4 7 2

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

3
2 5 4 3 6 1 0 7

No

No sequence of operations achieves the objective.

3
0 1 2 3 4 5 6 7

Yes
0

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