Bull in a China Shop timer 5s memory 512MB

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.

Problem credits: Brian Dean