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 io::*;

fn main() {
    let con = read_string();
    let mut sc = Scanner::new(&con);

    let n: usize = sc.next();
    let bd: Vec<Vec<char>> = (0..n)
        .map(|_| sc.next::<String>().chars().collect())
        .collect();

    let mut all_same = true;
    'outer: for i in 0..n {
        for j in 0..n {
            if bd[0][0] != bd[i][j] {
                all_same = false;
                break 'outer;
            }
        }
    }

    if all_same {
        println!("{}", n * n);
        return;
    }

    let mut nbd: Vec<Vec<char>> = vec![vec![' '; 2 * n]; 2 * n];
    for i in 0..2 * n {
        for j in 0..2 * n {
            nbd[i][j] = bd[j % n][i % n];
        }
    }

    let mut rbd: Vec<Vec<char>> = vec![vec![' '; 2 * n]; 2 * n];

    for i in 0..2 * n {
        for j in 0..2 * n {
            rbd[i][j] = nbd[j][i];
        }
    }

    let mut ans = 0;
    for a in 0..n {
        for b in 0..n {
            let mut ok = true;
            for i in 0..n {
                let r1 = &nbd[(i + a) % n];
                let r2 = &rbd[(i + b) % n];

                if &r1[b..b + i] != &r2[a..a + i] {
                    ok = false;
                    break;
                }

                // unsafe {
                //     for j in 0..i {
                //         if *r1.get_unchecked(j + b) != *r2.get_unchecked(j + a) {
                //             ok = false;
                //         }
                //     }
                // }
            }

            if ok {
                ans += 1;
            }
        }
    }

    println!("{:?}", ans);
}

//-----

pub mod io {
    use std;

    pub struct Scanner<'a> {
        iter: std::str::SplitWhitespace<'a>,
    }

    impl<'a> Scanner<'a> {
        pub fn new(s: &'a str) -> Scanner<'a> {
            Scanner {
                iter: s.split_whitespace(),
            }
        }

        pub fn next<T>(&mut self) -> T
        where
            T: std::str::FromStr,
            <T as std::str::FromStr>::Err: std::fmt::Debug,
        {
            let s = self.iter.next().unwrap();
            s.parse::<T>().unwrap()
        }
    }

    pub fn read_string() -> String {
        use std::io::Read;

        let mut s = String::new();
        std::io::stdin().read_to_string(&mut s).unwrap();
        s
    }

    pub fn read_line() -> String {
        let mut s = String::new();
        std::io::stdin().read_line(&mut s).unwrap();
        s.trim_right().to_owned()
    }
}