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++14 (GCC 5.4.1) (judge verdict: Runtime Error)

#include <bits/stdc++.h>
using namespace std;
const int N=305;

int n,ans;
char a[N][N];
long long h[N][N],v[N][N],p[N];

long long H(int i, int j, int k){
	return h[i+k][j+n-1]-h[i+k][j+k]*p[n-k-1];
}

long long V(int i, int j, int k){
	return v[i+n-1][j+k]-v[i+k][j+k]*p[n-k-1];
}

int main(){
	ios::sync_with_stdio(0);
	cin.tie(0);
	p[0]=1;
	for (int i=1;i<N;i++){
		p[i]=p[i-1]*42073;
	}
	cin >> n;
	for (int i=1;i<=n;i++){
		for (int j=1;j<=n;j++){
			cin >> a[i][j];
			a[i][j+n]=a[i+n][j]=a[i+n][j+n]=a[i][j];
		}
	}
	for (int i=1;i<=n*2;i++){
		for (int j=1;j<=n*2;j++){
			h[i][j]=h[i][j-1]*42073+a[i][j];
		}
	}
	for (int i=1;i<=n*2;i++){
		for (int j=1;j<=n*2;j++){
			v[j][i]=v[j-1][i]*42073+a[j][i];
		}
	}
	for (int i=1;i<=n;i++){
		for (int j=1;j<=n;j++){
			int ok=1;
			for (int k=0;k<n;k++){
				ok&=H(i,j,k)==V(i,j,k);
			}
			ans+=ok;
		}
	}
	cout << ans << "\n";
}