
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 the two foci of the hyperbola $x^2-\frac{y^2}{8}=1$ be $F_1$ and $F_2$, and let $P$ be a point on the hyperbola such that the ratio $|PF_1|:|PF_2|=3:4$. Calculate the area of $\triangle PF_1F_2$.

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