
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 the hyperbola \( C: x^{2} - \frac{y}{3} = 1 \), where the left and right foci are \( F_{1} \) and \( F_{2} \) respectively. A line passing through point \( F_{2} \) intersects the right branch of the hyperbola \( C \) at points \( P \) and \( Q \), such that \( \angle F_{1} P Q = 90^{\circ} \). Determine the radius of the inscribed circle of \( \triangle F_{1} P Q \).

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