A11368 | Cyclic Cipher

Senor Vorpal Kickass'o invented an innovative method to encrypt integer sequences of length $n$ . To encrypt a sequence, one has to choose a secret sequence , that acts as a key.

Vorpal is very selective, so the key should be such a sequence $b_{i}$ , that its cyclic shifts are linearly independent, that is, there is no non-zero set of coefficients $x_{0},x_{1},...,x_{n-1}$ , such that  for all $k$ at the same time.

After that for a sequence  you should build the following cipher:

In other words, you are to compute the quadratic deviation between each cyclic shift of $b_{i}$ and the sequence $a_{i}$ . The resulting sequence is the Kickass's cipher. The cipher is in development right now and Vorpal wants to decipher a sequence after it has been encrypted. You are to solve this problem for him. You are given sequences $c_{i}$ and $b_{i}$ . You are to find all suitable sequences $a_{i}$ .

The first line contains a single integer $n$ ().

The second line contains $n$ integers $b_{0},b_{1},...,b_{n-1}$ ().

The third line contains $n$ integers $c_{0},c_{1},...,c_{n-1}$ ().

It is guaranteed that all cyclic shifts of sequence $b_{i}$ are linearly independent.

In the first line print a single integer $k$ — the number of sequences $a_{i}$ , such that after encrypting them with key $b_{i}$ you get the sequence $c_{i}$ .

After that in each of $k$ next lines print $n$ integers $a_{0},a_{1},...,a_{n-1}$ . Print the sequences in lexicographical order.

Note that $k$ could be equal to $0$ .