
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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  $\triangle XYZ$ ?

Present the answer in LaTex format: \boxed{Your answer}
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