A973 | Replication--Gold

The ill-fated result of watching too many "do it yourself" engineering videos
on the web, Farmer John has accidentally released a self-replicating robot on
his farm!
The farm can be represented by an $N\times N$ grid ($3\le N\le 1000$) where
each grid cell is either empty or filled with rock, and all border squares are
filled with rock. Some non-rock cells are designated as possible starting
locations for the robot.
Farmer John initially places the robot at one of the possible starting
positions. In every hour that follows, all copies of the robot move in one
coordinated mass in the same direction, either north, south, east, or west.
After every $D$ hours ($1 \leq D \leq 10^9$), every copy of the robot
replicates \--- a robot at cell $(x,y)$ that replicates creates new copies in
cells $(x+1,y)$, $(x-1,y)$, $(x,y+1)$, and $(x,y-1)$; the original robot
remains at $(x,y)$. Over time, multiple robots might come to occupy the same
cell.
If moving or replicating would cause any of the robots to move into a rock,
then all robots shut down immediately. Note that this implies that the robots
must eventually shut down, due to the border of the farm being rock.
Help the cows figure out the number of empty squares that could potentially at
some point in time hold a robot.

The first line contains two space-separated integers $N$ and $D$. The next $N$
lines of input each contain $N$ characters. Each character is one of '.', 'S',