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 practice1
{
    class Program
    {
        static void Main(string[] args)
        {
            char[,] field1, field2;
            int count, i, j, n;
            count = 0;
            string buff;

            n = int.Parse(Console.ReadLine());

            field1 = new char[n, n];

            for (i = 0; i < n; i++)
            {
                buff = Console.ReadLine();

                for (j = 0; j < n; j++)
                {
                    field1[i, j] = buff.ToCharArray()[j];
                }
            }

            for (i = 0; i < n; i++)
            {
                for (j = 0; j < n; j++)
                {
                    field2 = MakeField(field1, n, i, j);
                    if (CheckEqual(field2, n)) count++;
                    field2 = null;
                }
            }

            MYCP.PrintNumber(count);

            Console.ReadLine();
        }

        public static char[,] MakeField(char[,] field, int n, int a, int b)
        {
            char[,] result = new char[n, n];
            int i, j;

            for (i = 0; i < n; i++)
            {
                for (j = 0; j < n; j++)
                {
                    result[i, j] = field[(i + a) % n, (j + b) % n];
                }
            }

            return result;
        }

        public static bool CheckEqual(char[,] a, int n)
        {
            int i, j;

            for (i = 0; i < n; i++)
            {
                for (j = 0; j < n; j++)
                {
                    if (a[i, j] != a[j, i]) return false;
                }
            }

            return true;
        }

    }

    

    class MYCP
    {
        
        


        //int変数を標準出力する
        public static void PrintNumber(long number)
        {
            Console.WriteLine("" + number);
        }

        //int変数の配列を標準出力する
        public static void PrintNumbers(int[] a)
        {
            if (a.Length == 0) return;

            Console.Write(a[0]);

            for(int i = 1; i < a.Length; i++)
            {
                Console.Write(" ");
                Console.Write(a[i]);
            }

            Console.WriteLine();
        }


        //文字列を配列化して返す
        public static string[] MyReadLineSplit()
        {
            return Console.ReadLine().Split();
        }


        //double変数をソートする
        public static void MyDoubleSort(double[] boo)
        {
            if (boo.Length < 2) return;

            long i, count;
            double a;

            while (true)
            {
                count = 0;

                for (i = 0; i < boo.Length - 1; i++)
                {
                    if (boo[i] > boo[i + 1])
                    {
                        a = boo[i];
                        boo[i] = boo[i + 1];
                        boo[i + 1] = a;
                        count++;
                    }
                }

                if (count == 0) break;
            }
        }

        //int変数をソートする
        public static void MyIntSort(int[] boo)
        {
            if (boo.Length < 2) return;

            long i, count;
            int a;

            while (true)
            {
                count = 0;

                for (i = 0; i < boo.Length - 1; i++)
                {
                    if (boo[i] > boo[i + 1])
                    {
                        a = boo[i];
                        boo[i] = boo[i + 1];
                        boo[i + 1] = a;
                        count++;
                    }
                }

                if (count == 0) break;
            }
        }

        //指定された個数分int変数を読み取る
        public static int[] ReadNumbers(long number)
        {
            int[] a = new int[number];
            string[] buff = MyReadLineSplit();
            for (long i = 0; i < number; i++)
            {
                a[i] = int.Parse(buff[i]);
            }

            return a;
        }

        //指定された個数分BigInteger変数を読み取る
        public static BigInteger[] ReadBigIntegers(long number)
        {
            BigInteger[] a = new BigInteger[number];
            string[] buff = MyReadLineSplit();
            for(long i = 0; i < number; i++)
            {
                a[i] = BigInteger.Parse(buff[i]);
            }

            return a;
        }



        //渡された自然数が素数か判定する。素数ならtrueを返す。
        public static bool PrimeCheck(int number)
        {
            if (number < 2) return false;

            for(int i = 2; i < number; i++)
            {
                if (number % i == 0) return false;
            }

            return true;
        }

        //渡された自然数が累乗数か判定する。累乗数ならtrueを返す。
        public static bool Bekizyousuu_Check(int number)
        {
            if (number == 1) return true;

            int i, copy;


            for (i = 2; i < number; i++)
            {
                copy = number;
                while (copy % i == 0)
                {
                    copy /= i;
                }

                if (copy == 1) return true;
            }

            return false;
        }

        //BIgIntegerで累乗計算を行う
        public static BigInteger MyPowerBigInteger(BigInteger a,BigInteger b)
        {
            BigInteger i, x = 1;

            for (i = 0; i < b; i++)
            {
                x *= a;
            }

            return x;
        }

        //渡されたBigIntegerの配列の和を求める
        public static BigInteger MySumBigInteger(BigInteger[] a)
        {
            BigInteger sum = 0;
            int i;

            for (i = 0; i < a.Length; i++)
            {
                sum += a[i];
            }

            return sum;

        }
    }
}