Problem Statement

Rusk has a sequence $A=(A_0, A_1, \ldots, A_{2^N-1})$ of length $2^N$. Each element of $A$ is either $-1$ or an integer between $0$ and less than $2^{30}$. Rusk wants to replace every $-1$ contained in $A$ with an integer that is at least $0$ and less than $2^{30}$ so that the following condition is satisfied.

• For all $i$, $j$ ($0\leq i, j < 2^N$), $A_{i\oplus j}\geq A_i\oplus A_j$ holds.

Here, for non-negative integers $x$, $y$, $x\oplus y$ denotes the bitwise XOR of $x$ and $y$.

Input

$N$
$A_0\ A_1\ \ldots \ A_{2^N-1}$

• $1\leq N\leq 7$
• $-1\leq A_i < 2^{30}$
• All input values are integers.

Output

If no replacement satisfying the condition exists, print No. Otherwise, print Yes on line $1$, and on line $2$, print the elements of $A$ after a replacement satisfying the condition, separated by spaces. If there are multiple valid ways of replacing, you may print any of them.

Sample

2
2 -1 -1 6

Yes
2 7 3 6

3
-1 4 -1 5 9 2 6 5

No

3
3 -1 -1 7 -1 -1 -1 10

Yes
3 22 18 7 15 26 30 10