
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
Given a circle of radius $r$, chords $\overline{AB}$ of length $12$ and $\overline{CD}$ of length $9$ are extended through $B$ and $C$, intersecting at point $P$ outside the circle, where $\angle{APD}=60^\circ$ and $BP=10$. Find the value of $r^2$.

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