
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 $F_1$ and $F_2$ be the left and right foci of the hyperbola $C:x^2-\frac{y^2}{24}=1$, respectively. Let $P$ be a point on the hyperbola $C$ in the first quadrant. If $\frac{|PF_1|}{|PF_2|}=\frac{4}{3}$, then the area of the inscribed circle of $\triangle PF_1F_2$ is ______.

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