
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 hyperbola centered at the origin with its foci on the x-axis and a focal distance of 12, a circle defined by $(x-6)^{2}+y^{2}=20$ is tangent to the asymptote of the hyperbola. Point $P$ lies on the hyperbola, and the distance from $P$ to one focus, $F_1$, is 9. Determine the distance from point $P$ to the other focus, $F_2$.

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