
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
Triangle  $ABC$  is a triangle with side lengths  $13$ ,  $14$ , and  $15$ . A point  $Q$  is chosen uniformly at random in the interior of  $\triangle{ABC}$ . Choose a random ray (of uniformly random direction) with endpoint  $Q$  and let it intersect the perimeter of  $\triangle{ABC}$  at  $P$ . What is the expected value of  $QP^2$ ?

*2018 CCA Math Bonanza Tiebreaker Round #4*

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