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