A891.262144--Platinum

Farmer John has decided his home needs more decoration. Visiting the local
china shop, he finds a delicate glass cow figurine that he decides to
purchase, knowing that it will fit perfectly on the mantel above his
fireplace.
The shape of the cow figurine is described by an $N \times M$ grid of
characters like the one below ($3 \leq N, M \leq 500$), where lowercase letter
characters are each part of the figurine (indicating different colors) and '.'
characters are not.
...............
...............
x..x...........
xxxx...........
xxxxaaaaaaa...
.xx.aaaaaaaaa..
....aaaaaaa.aa.
....ll...ll....
....vv...vv....
...............
Unfortunately, right before FJ can make his purchase, a bull runs through the
shop and breaks not only FJ's figurine, but many of the other glass objects on
the shelves as well! FJ's figurine breaks into 3 pieces, which quickly become
lost among $K$ total pieces lying on the ground ($4 \leq K \leq 100$). Each of
the $K$ pieces is described by a grid of characters, just like the original
figurine.
Please help FJ determine how many sets of 3 pieces (out of the $K$ on the
floor) could be glued back together to mend his broken figurine.
The pieces on the ground might have been flipped vertically or horizontally,
or rotated by some multiple of 90 degrees. Therefore, given the original grid
as well as $K$ grids describing pieces, you want to find sets of 3 pieces that
can be joined together to form the original picture, allowing the pieces to be
translated, flipped, or rotated multiples of 90 degrees. When then
superimposed, the 3 pieces should exactly form the original picture, with each
colored square in the original picture represented in exactly one of the
pieces.

The first line contains a single integer $K$. Following that will be $K + 1$
piece descriptions. The first description will describe the original glass
cow, the following $K$ descriptions will be of the broken pieces.
Each description begins with a line containing two integers $R$ and $C$ ($1 \le R, C \le 100$). The following $R$ lines contain $C$ lowercase alphabet
characters describing the color of each cell. Each piece will be
horizontally/vertically connected and have at least one non-empty cell.

Output the number of triples $i, j, k$ ($i < j < k$) such that pieces $i$,
$j$, and $k$ can be arranged to form the original glass cow.

The three solutions use pieces $(0, 1, 2)$, $(0, 2, 4)$, $(1, 3, 4)$.
Note that this problem has a time limit of 6 seconds per test case (and twice
that for Java and Python submissions).