Let $H$ be a regular hexagon with area 360. Three distinct vertices $X$, $Y$, and $Z$ are picked randomly, with all possible triples of distinct vertices equally likely. Let $A$, $B$, and $C$ be the unpicked vertices. What is the expected value (average value) of the area of the intersection of $\triangle ABC$ and $\triangle XYZ$?

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