Problem:
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.

Solution in Rust (1.15.1) (judge verdict: Time Limit Exceeded)

use std::io::BufRead;

fn main() {
    let stdin = std::io::stdin();
    let n = stdin.lock().gets().parse::<usize>().unwrap();
    let map = stdin.lock().lines().take(n).map(|x| x.unwrap().into_bytes())
        .collect::<Vec<_>>();
    let mut starts = Vec::new();
    for x in 0..n {
        starts.push((x, 0));
    }
    for y in 1..n {
        starts.push((0, y));
    }
    let mut ans = 0u64;
    for (mut sx, mut sy) in starts {
        let mut symcount = Vec::new();
        while sx < 2 * n - 1 && sy < 2 * n - 1 {
            let mut count = 0;
            for i in 1..n {
                if map[sy % n][(sx + i) % n] != map[(sy + i) % n][sx % n] {
                    break;
                }
                count += 1;
            }
            symcount.push(count);
            sx += 1;
            sy += 1;
        }
        'notsym: for i in 0..(symcount.len() - n + 1) {
            for j in 0..(n - 1) {
                if symcount[i + j] < n - j - 1 {
                    continue 'notsym;
                }
            }
            ans += 1;
        }
    }
    println!("{}", ans);
}

trait Gets { fn gets(&mut self) -> String; }
impl<T: std::io::BufRead> Gets for T {
    fn gets(&mut self) -> String { self.lines().next().unwrap().unwrap() } }