Problem Statement

Given are sequences $a$ and $b$, each of length $N$, consisting of non-negative integers. The $i$-th elements of $a$ and $b$ are $a_i$ and $b_i$, respectively.

A non-negative integer $x$ is said to be good when the following condition is satisfied:

• Condition: It is possible to permute $b$ so that $a_i \text{ XOR } b_i = x$ holds for every integer $i$ such that $1 \leq i \leq N$, where $\text{XOR }$ is the bitwise XOR.

List all good integers in ascending order.

What is $\mathrm{XOR}$? The bitwise $\mathrm{XOR}$ of integers $x$ and $y$, $x\ \mathrm{XOR}\ y$, is defined as follows: When $x\ \mathrm{XOR}\ y$ is written in base two, the digit in the $2^k$'s place ($k \geq 0$) is $1$ if exactly one of $x$ and $y$ is $1$, and $0$ otherwise. For example, we have $3\ \mathrm{XOR}\ 5 = 6$ (in base two: $011\ \mathrm{XOR}\ 101 = 110$).

Constraints

• All values in input are integers.
• $1 \leq N \leq 2000$
• $0 \leq a_i, b_i < 2^{30}$

Input

Input is given from Standard Input in the following format:

N
a_1 \cdots a_N
b_1 \cdots b_N

Output

In the first line, print $K$, the number of good integers. Then, print $K$ more lines. In the $i$-th of these $K$ lines, print the $i$-th smallest good integer.