4 \(s / 256\) \(MB / 160\) points

We have created an infinite eig\(ht-di\)rectional crossword by taking a lett\(er-fi\)lled block of

dimensions \(M\)

x \(N\)

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 inifinite 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.

In test cases worth 100 total points, it will hold \(M\)

= \(N\)

.

The first line of input contains integers M,

N, K

from the task (1 ≤ M, N

≤ 500, 2 ≤ \(K\)

≤ 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.

You must output the required probability in the form of a reduced fraction \(p/q\),

without
spaces.

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출처 COCI 2016/2017 Contest 4