Three squares surround an equilateral triangle with a side length $2$, each one sharing an edge with the triangle, as shown in the figure below. The area of the resulting 9-sided outer nonconvex polygon can be written as $m\sqrt{n} + p$, where $m$, $n$, and $p$ are integers and $n$ is not divisible by the square of any prime. What is $m + n + p$?

1. The three outward squares each have area $2^2=4$, so their total area is $12$. Their interiors do not overlap each other or the central triangle.
2. The central equilateral triangle has area $\dfrac{\sqrt3}{4}(2^2)=\sqrt3$. Thus the $9$-sided region has area $12+\sqrt3$.
3. $(m,n,p)=(1,3,12)$, so $m+n+p=\boxed{16}$.