Question 12
Let $ABCD$ be a parallelogram with $\angle BAD \lt 90^\circ$. A circle tangent to sides $DA$, $AB$, and $BC$ intersects diagonal $AC$ at points $P$ and $Q$ with $AP \lt AQ$, as shown. Suppose that $AP = 3$, $PQ = 9$, and $QC = 16$. Then the area of $ABCD$ can be expressed in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.
