﻿ AGC023 B - Find Symmetries - Atcoder

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Score : $500$ points

### Problem Statement

Snuke has two boards, each divided into a grid with $N$ rows and $N$ columns. For both of these boards, the square at the $i$-th row from the top and the $j$-th column from the left is called Square $(i,j)$.

There is a lowercase English letter written in each square on the first board. The letter written in Square $(i,j)$ is $S_{i,j}$. On the second board, nothing is written yet.

Snuke will write letters on the second board, as follows:

• First, choose two integers $A$ and $B$ ( $0 \leq A, B < N$ ).
• Write one letter in each square on the second board. Specifically, write the letter written in Square $( i+A, j+B )$ on the first board into Square $(i,j)$ on the second board. Here, the $k$-th row is also represented as the $(N+k)$-th row, and the $k$-th column is also represented as the $(N+k)$-th column.

After this operation, the second board is called a good board when, for every $i$ and $j$ ( $1 \leq i, j \leq N$ ), the letter in Square $(i,j)$ and the letter in Square $(j,i)$ are equal.

Find the number of the ways to choose integers $A$ and $B$ ( $0 \leq A, B < N$ ) such that the second board is a good board.

### Constraints

• $1 \leq N \leq 300$
• $S_{i,j}$ ( $1 \leq i, j \leq N$ ) is a lowercase English letter.

### Input

Input is given from Standard Input in the following format:

$N$
$S_{1,1}S_{1,2}..S_{1,N}$
$S_{2,1}S_{2,2}..S_{2,N}$
$:$
$S_{N,1}S_{N,2}..S_{N,N}$


### Output

Print the number of the ways to choose integers $A$ and $B$ ( $0 \leq A, B < N$ ) such that the second board is a good board.

### Sample Input 1

2
ab
ca


### Sample Output 1

2


For each pair of $A$ and $B$, the second board will look as shown below:

The second board is a good board when $(A,B) = (0,1)$ or $(A,B) = (1,0)$, thus the answer is $2$.

### Sample Input 2

4
aaaa
aaaa
aaaa
aaaa


### Sample Output 2

16


Every possible choice of $A$ and $B$ makes the second board good.

### Sample Input 3

5
abcde
fghij
klmno
pqrst
uvwxy


### Sample Output 3

0


No possible choice of $A$ and $B$ makes the second board good.