Artificial Strife

Conway's Game of Life is a simulation run on a square grid of cells. Each cell is either alive or dead, and each turn of the simulation is based on the contents of the previous turn; that is, each cell's state in turn n+1 is based on its state in turn n along with the states of the eight cells surrounding it. In the standard Game of Life, a cell remains alive if it has two or three neighbours, and an empty cell becomes alive if it has precisely three neighbours: otherwise, a cell dies or stays empty. The figure below shows three generations of a simple but surprisingly complex Conwayian structure called a glider:

   0       1       2       3       4....... ....... ....... ....... .............. ...A... ..AA... ..AA... .AAA.....AAA.. ..AA... ..A.A.. .AA.... .A.......A.... ..A.A.. ..A.... ...A... ..A.......A... ....... ....... ....... .............. ....... ....... ....... .......

As one can see, the entire structure has in fact moved one cell up and one cell to the left after four turns. There are many other much more complex structures in the Game of Life; it is even possible to build a (very, very slow) Turing Machine on the grid, or simulate the Game of Life itself!

The ruleset for the original Game of Life can be described as 23/3; that is, a cell stays alive in turn n+1 if it has two or three live neighbours in turn n, and a dead cell turns live if it has precisely three neighbours. There are many other "Conway-like" rulesets that can be described with the same notation. Note that a ruleset like /234 (that is, all currently live cells die, but any dead cells with two, three, or four neighbors come alive) is perfectly valid (and quite interesting); rulesets with no "birth values" are also valid but (unsurprisingly) dull. Your goal is to simulate a number of these rules...at the same time on the same grid.

Because there is more than one ruleset running on the same grid, a number of clarifications are required:

• Each cell can be occupied by at most one cell from one species (a ruleset represented by a particular letter);

• Only cells of the same species count towards the state of the next turn for that species;

• When there is a conflict as to the state of a cell in the next turn, resolution is as follows:

  1. If one ruleset will place a live cell and all others leave the cell dead, place a live cell of the ruleset that allows it;
  2. If more than one ruleset will place a live cell in that location, the ruleset whose representation is first in alphabetical order wins (that is, if species B and D would both have a live cell in a location, the cell becomes a live cell with species B.)

• One can assume that the board is surrounded by an infinite number of dead cells that cannot become alive.

After running a simulation for a given number of turns, you are to report the maximum and minimum population of each species involved.

Input to this problem will begin with a line containing a single integer n indicating the number of simulations in the file. For each simulation, the first line contains three integers X Y S (1 <= X,Y <= 50; 1 <= S <= 26) where X and Y represent the width and height of the board and S represents the number of different species represented on the board. The next Y lines are a textual representation of the board in turn 0, as shown above. The following S lines are the definitions of the rulesets for each species, given in the format described above. The last line of each data set is an integer T representing the number of turns to simulate. (For example, the figure above was a simulation of 4 turns.)

For each data set, first print "Simulation #N" where N is the number of the data set, starting at 1. Then print S lines, each of the format "Species S: At most M live, at least L live." where S is the letter representation of the species (starting with A), M is the maximum number of that species alive in any turn of the simulation (including turn 0), and L is the least number of that species alive in any turn of the simulation (including turn 0).

The below diagram shows the initial state plus the four following states of the second sample input:

  0     1     2     3     4..A.. ..... ..A.. ..... ..A....A.. .AAA. ..A.. .AAA. ..A....A.. ..... ..A.. ..B.. ..A....... ..B.. .BBB. .B.B. .B.B..BBB. .BBB. .BBB. .B.B. .B.B.

Note that in turn 4 the A species takes over a cell that would have otherwise stayed as a B.