CF901E.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.

森奥·沃帕尔·基卡索（Senor Vorpal Kickass'o）发明了一种创新方法，用于加密长度为 $n$ 的整数序列。要加密一个序列，需选择一个作为密钥的秘密序列 。

沃帕尔非常挑剔，因此该密钥序列 $b_i$ 必须满足：其所有循环移位是线性无关的，即不存在一组不全为零的系数 $x_0,\,x_1,\,...,\,x_{n-1}$，使得对所有 $k$ 同时成立
。

随后，对于给定序列 ，应构造如下密文：

换言之，需对 $b_i$ 的每个循环移位与序列 $a_i$ 计算二次偏差（即平方差之和）。所得序列即为基卡索密文（Kickass's cipher）。该密文目前尚在开发中，而沃帕尔希望在序列被加密后能将其解密。你需要帮他解决这一问题。现给出密文序列 $c_i$ 和密钥序列 $b_i$，请找出所有满足条件的原序列 $a_i$。

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.

第一行包含一个整数 $ n $（）。

第二行包含 $ n $ 个整数 $ b_0,,b_1,,\dots,,b_{n-1} $（）。

第三行包含 $ n $ 个整数 $ c_0,,c_1,,\dots,,c_{n-1} $（）。

保证序列 $ b_i $ 的所有循环移位是线性无关的。

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.

第一行输出一个整数 $k$ —— 满足“使用密钥 $b_i$ 对其加密后得到序列 $c_i$”的序列 $a_i$ 的个数。

随后的 $k$ 行中，每行输出 $n$ 个整数 $a_0,\ a_1,\ \ldots,\ a_{n-1}$。请按字典序输出这些序列。

注意：$k$ 可能为 0。