A996.Paint by Letters--Platinum

Bessie has recently received a painting set. The canvas can be represented as
an $N \times M$ rectangle of cells where the rows are labeled $1\ldots N$ from
top to bottom and the columns are labeled $1\ldots M$ from left to right
($1\le N,M\le 1000$). Once painted, the color of a cell can be represented by
an uppercase letter from 'A' to 'Z.' Initially, all cells are uncolored, and a
cell cannot be painted more than once.
Bessie has specified the color that she desires for each cell. She can paint a
set of cells with a single color in one stroke if the set forms a connected
component, meaning that any cell in the set can reach any other via a sequence
of adjacent cells. Two cells are considered to be adjacent if they share an
edge.
For example, the $3\times 3$ canvas
AAB
BBA
BBB
can be colored in four strokes as follows:
... ..B AAB AAB AAB
... -> ... -> ... -> BB. -> BBA
... ... ... BBB BBB
It is not possible to produce the end result using less than four strokes.
Being an avant-garde artist, Bessie will end up painting only a subrectangle
of the canvas. Currently, she is considering $Q$ candidates ($1\le Q\le 1000$), each of which can be represented by four integers $x_1$, $y_1$, $x_2$,
and $y_2.$ This means that the subrectangle consists of all cells with row in
the range $x_1$ to $x_2$ inclusive and column in the range $y_1$ to $y_2$
inclusive.
For each candidate subrectangle, what is the minimum number of strokes needed
to paint each cell in the subrectangle with its desired color while leaving
all cells outside the subrectangle uncolored? Note that Bessie does not
actually do any painting during this process, so the answers for each
candidate are independent.
Note: The time limit for this problem is 50 percent higher than the default,
and the memory limit is 512MB, twice the default.

The first line contains $N$, $M$, and $Q$.
The next $N$ lines each contain a string of $M$ uppercase characters
representing the desired colors for each row of the canvas.
The next $Q$ lines each contain four space-separated integers
$x_1,y_1,x_2,y_2$ representing a candidate subrectangle ($1\le x_1\le x_2\le N$, $1\le y_1\le y_2\le M$).

For each of the $Q$ candidates, output the answer on a new line.

The first candidate consists of the entire canvas, which can be painted in six
strokes.
The second candidate consists of the subrectangle with desired colors
ABBA
and can be colored in three strokes. Note that although the cells at $(3,5)$
and $(3,8)$ can be colored with $A$ in a single stroke if you consider the
entire canvas, this is not the case when considering only the cells within the
subrectangle.