Question 21
Let $ABCDEF$ be an equiangular hexagon. The lines $AB,$ $CD,$ and $EF$ determine a triangle with area $192\sqrt{3},$ and the lines $BC,$ $DE,$ and $FA$ determine a triangle with area $324\sqrt{3}.$ The perimeter of hexagon $ABCDEF$ can be expressed as $m +n\sqrt{p},$ where $m,$ $n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. What is $m + n + p?$