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 Python (3.4.3) (judge verdict: Time Limit Exceeded)

from sys import exit, setrecursionlimit
from functools import reduce
from itertools import *
from collections import defaultdict
from bisect import bisect

def read():
  return int(input())

def reads():
  return [int(x) for x in input().split()]

p = 10**9 + 7
M = 1 << 64

N = read()
S = []

for _ in range(N):
  S.append([ord(c) for c in input()])

St = [[S[j][i] for j in range(N)] for i in range(N)]

rhS = [[0] * (N+1) for i in range(N)]
for i in range(N):
  for j in range(N):
    rhS[i][j+1] = (rhS[i][j] * p + S[i][j]) % M

rhSt = [[0] * (N+1) for i in range(N)]
for i in range(N):
  for j in range(N):
    rhSt[i][j+1] = (rhSt[i][j] * p + St[i][j]) % M

pp = [1] * (N+11)
for k in range(N+10):
  pp[k+1] = (pp[k] * p) % M

def rotate(rhRow, k):
  return (rhRow[k] + (rhRow[N] - rhRow[k] * pp[N-k]) * pp[k]) & (M-1)

ans = 0
for A in range(N):
  for B in range(N):
    if all(rotate(rhS[(i + A) % N], B) == rotate(rhSt[(i + B) % N], A) for i in range(N)):
      ans += 1

print(ans)