
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, and let \( P \) be a point on the hyperbola \( C \) in the first quadrant. If \( \frac{\left|P F_{1}\right|}{\left|P F_{2}\right|} = \frac{4}{3} \), find the radius of the inscribed circle of the triangle \( P F_{1} F_{2} \).

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