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 C# (Mono 4.6.2.0) (judge verdict: Time Limit Exceeded)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
using System.Numerics;

namespace AtCoder
{
    class Program
    {
        static void Main(string[] args)
        {
            var N = int.Parse(Console.ReadLine());
            var board = new char[N][];
            foreach (var i in Enumerable.Range(0, N))
            {
                board[i] = Console.ReadLine().ToCharArray();
            }

            var result = 0;
            foreach(var a in Enumerable.Range(0, N))
            {
                foreach(var b in Enumerable.Range(0, N))
                {
                    var target = MakeBoard(a, b, board, N);
                    if (IsGoodBoard(target, N)) result++;
                }
            }

            Console.WriteLine(result);
        }

        static char[,] MakeBoard(int A, int B, char[][] board, int N)
        {
            var result = new char[N, N];
            foreach(var i in Enumerable.Range(0, N))
            {
                foreach(var j in Enumerable.Range(0, N))
                {
                    result[(i + A) % N, (j + B) % N] = board[i][j];
                }
            }
            return result;
        }

        static bool IsGoodBoard(char[,] targetBoard, int N)
        {
            foreach(var i in Enumerable.Range(0, N))
            {
                for(var j=i+1; j<N; j++)
                {
                    if (targetBoard[i, j] != targetBoard[j, i]) return false;
                }
            }
            return true;
        }
    }
}