Four regular hexagons surround a square with a side length $1$, each one sharing an edge with the square, as shown in the figure. The area of the regions where the hexagons overlap each other (excluding the central square) 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 $m+n+p$?