Four regular hexagons surround a square with a side length $2$, each one sharing an edge with the square, in a similar configuration to the figure shown. The area of the resulting $12$-sided outer nonconvex polygon can be written as $m\sqrt{n}+p$, where $m,n,p$ are integers and $n$ is not divisible by the square of any prime. What is $m+n+p$?