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.