We have created an infinite eight-directional crossword by taking a letter-filled block of dimensions ~M \times N~ and infinitely repeating it. For instance, if we are given the following block:

honi
hsin

then we create the following crossword:

...honihonihonihoni...
...hsinhsinhsinhsin...
...honihonihonihoni...
...hsinhsinhsinhsin...

that is infinite in all directions.

In the created crossword, we randomly choose a field and one of eight directions, then write down a word of length ~K~ obtained by reading the crossword starting from the initial field, in the chosen direction. If we executed this query twice (independently), we would obtain two words of length ~K~. Calculate the probability that the two words are equal.

Input Specification

The first line of input contains integers ~M~, ~N~, ~K~ from the task ~(1 \le M, N \le 500, 2 \le K \le 10^9)~.

Each of the following ~M~ lines contains ~N~ lowercase letters of the English alphabet, and describes a block of the crossword. At least two distinct letters will exist in the block.

Output Specification

You must output the required probability in the form of a reduced fraction ~p/q~, without spaces.

Scoring

In test cases worth ~100~ total points, it will hold ~M = N~.

Sample Input 1

1 2 2
ab

Sample Output 1

5/16

Sample Input 2

2 4 3
honi
hsin

Sample Output 2

19/512

Sample Input 3

3 3 10
ban
ana
nab

Sample Output 3

2/27