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 (Clang 3.8.0) (judge verdict: Time Limit Exceeded)

#include <stdbool.h>
#include <stdio.h>
#include <stdlib.h>
#include <inttypes.h>
#include <stdint.h>
#include <string.h>

char first[300][300];

bool isGoodBoard(const uint32_t N, const uint32_t A, const uint32_t B) {
	for (uint32_t i = 0; i < N; i++) {
		for (uint32_t j = i; j < N; j++) {
			if (first[(A + i) % N][(B + j) % N] != first[(A + j) % N][(B + i) % N]) {
				return false;
			}
		}
	}

	return true;
}

int main(void) {
	uint32_t N;
	uint32_t goodCount = 0;

	scanf("%"PRIu32"\n", &N);
	for (uint32_t rowIdx = 0; rowIdx < N; rowIdx++) {
		for (uint32_t colIdx = 0; colIdx < N; colIdx++) {
			scanf("%c", &(first[rowIdx][colIdx]));
		}
		if (rowIdx != N-1)
			scanf("\n");
	}

	for (uint32_t A = 0; A < N; A++) {
		for (uint32_t B = 0; B < N; B++) {
			if (isGoodBoard(N, A, B)) {
				goodCount++;
			}
		}
	}

	printf("%"PRIu32, goodCount);

	return 0;
}