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 LuaJIT (2.0.4) (judge verdict: Time Limit Exceeded)

local n = io.read("*n", "*l")
local row = {}
for i = 1, n do
  row[i] = io.read()
end
local col = {}
for i = 1, n do col[i] = "" end
for i = 1, n do
  for j = 1, n do
    col[i] = col[i] .. row[j]:sub(i, i)
  end
end

local colmap = {}
for i = 1, n do
  local tmps = col[i]
  colmap[i] = {tmps}
  for j = 2, n do
    tmps = tmps:sub(n, n) .. tmps:sub(1, n - 1)
    colmap[i][j] = tmps
  end
end
local rowmap = {}
for i = 1, n do
  local tmps = row[i]
  rowmap[i] = {tmps}
  for j = 2, n do
    tmps = tmps:sub(n, n) .. tmps:sub(1, n - 1)
    rowmap[i][j] = tmps
  end
end

local function check(r, c)
  for k = 1, n do
    local row_tgt = k + n - r
    if n < row_tgt then row_tgt = row_tgt - n end
    local row_ofst = 1 + c
    local col_tgt = k + n - c
    if n < col_tgt then col_tgt = col_tgt - n end
    local col_ofst = 1 + r
    if rowmap[row_tgt][row_ofst] ~= colmap[col_tgt][col_ofst] then return 0 end
  end
  return 1
end

local ret = 0
for i = 0, n - 1 do
  for j = 0, n - 1 do
    ret = ret + check(i, j)
  end
end
print(ret)