Let $O$ be a point with three other points $A,B,C$ and $\angle AOB=\angle BOC=\angle AOC=2\pi/3$.  Consider the average area of the set of triangles $ABC$ where $OA,OB,OC\in\{3,4,5\}$.  The average area can be written in the form $m\sqrt n$ where $m,n$ are integers and $n$ is not divisible by a perfect square greater than $1$.  Find $m+n$.

Let's think step by step and output the final answer within \boxed{}.
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