A996. Sprinklers 2 Return of the Alfalfa

Farmer John has a small field in the shape of an $N$ by $N$ grid ($1 \le N \le
2000$) where the $j$-th square from the left of the $i$-th row from the top is
denoted by $(i,j)$ for all $1 \le i,j \le N$. He is interested in planting
sweet corn and alfalfa in his field. To do so, he needs to install some
special sprinklers.
A sweet corn sprinkler in square $(I,J)$ will sprinkle all squares to the
bottom-left: i.e. $(i,j)$ with $I \le i$ and $j \le J$.
An alfalfa sprinkler in square $(I,J)$ will sprinkle all squares to the top-
right: i.e. $(i,j)$ with $i \le I$ and $J \le j$.
A square sprinkled by one or multiple sweet corn sprinklers can grow sweet
corn; a square sprinkled by one or multiple alfalfa sprinklers can grow
alfalfa. But a square sprinkled by both types of sprinklers (or neither type)
can grow nothing.
Help FJ determine the number of ways (modulo $10^9 + 7$) to install sprinklers
in his field, at most one per square, so that every square is fertile (i.e.,
sprinkled by exactly one type of sprinkler).
Some of the squares are already occupied by woolly cows; this doesn't prevent
these squares from being fertile, but no sprinklers can be installed in such
squares.

The first line contains a single integer $N.$
For each $1\le i\le N,$ the $i+1$-st line contains a string of length $N$
denoting the $i$-th row of the grid. Each character of the string is one of
'W' (indicating a square occupied by a woolly cow), or '.' (unoccupied).

Output the remainder when the number of ways to install sprinklers is divided
by $10^9+7.$

Here are all fourteen possibilities when sweet corn can grow at $(1,1)$.
CC .C CA CC .C CA CA C. CA C. CC .C CC .C
CC, CC, CC, .C, .C, .C, CA, CA, .A, .A, C., C., .., ..