# Problem D

Vin Diagrams

Venn diagrams were invented by the great logician John Venn as a way of categorizing elements belonging to different sets. Given two sets $A$ and $B$, two overlapping circles are drawn – a circle representing the elements of $A$, and another representing the elements of $B$. The overlapping region of the circles represents element that belong to both $A$ and $B$, i.e., the intersection of the two sets $A \cap B$. A classic Venn diagram might look like this:

One of John’s biggest fans was his grandson, Vin Vaughn Venn. Vin was inspired by his grandfather’s diagrams, but Vin was a very creative individual. Simple overlapping circles struck Vin as too boring of a way to visualize the sometimes messy intersections of categories, so he set out to make his grandfather’s diagrams more interesting. Just like Venn diagrams, Vin diagrams are used as a way of categorizing elements belonging to different sets $A$ and $B$, but the representation of each set is not required to be a circle. In fact, each set can have any shape as long as there is single overlapping section for elements in the intersection of $A$ and $B$.

In this problem, Vin diagrams will be laid out on a grid.
Each set representation is a loop of ‘`X`’
characters, with one ‘`X`’ in each loop
replaced by an ‘`A`’ or ‘`B`’ to identify the loop. All empty positions (both
inside and outside of the loops) are represented by period
(‘.’) characters, and the set of positions inside a loop is
contiguous. Each loop character touches exactly two other loop
characters either vertically or horizontally. Loops do not
self-intersect, and other than the allowed horizontal/vertical
paths and right angle connections, different parts of the loop
do not touch (see Figures 2 and 3 below).

Loops A and B intersect at exactly two points. Loop intersection points always follow the pattern shown in Figure 4 (including the four ‘.’ positions around the intersection). No loop makes a right angle turn at an intersection point but always flows straight through the intersection, either vertically or horizontally. An example of legally intersecting loops is shown in Figure 5.

## Input

The input starts with two integers $r$ $c$ describing the number of rows and columns in the Vin diagram ($7 \leq r, c \leq 100$). The following $r$ rows each contain a string of $c$ characters. All positions that are not part of loop $A$ or loop $B$ are marked with a period (‘.’) character. The loop labels ‘A’ and ‘B’ are placed somewhere around the loops’ perimeters at non-intersection positions and are never on the same loop. The two loops will touch only at the two points where they intersect.

## Output

Display, in order, the area of the Vin diagram exclusive to set $A$, the area exclusive to set $B$, and the area of the intersection. Given the representation of Vin diagrams, the area of a section is defined as the number of periods (‘.’) it encloses.

Sample Input 1 | Sample Output 1 |
---|---|

7 7 AXXXX.. X...X.. X.XXXXX X.X.X.X XXXXX.X ..X...X ..XXXXB |
5 5 1 |

Sample Input 2 | Sample Output 2 |
---|---|

11 13 XXXXXXA...... X.....X...... X..XXXXXXXXX. X..X..X....X. X..X..XXX..XX X..B....X...X X..X.XXXX...X X..X.X......X XX.XXXXXX...X .X...X..X.XXX .XXXXX..XXX.. |
21 22 10 |