
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 \( C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \) (where \( a > 0 \) and \( b > 0 \)), its eccentricity is \( \frac{\sqrt{5}}{2} \). The points \( F_{1} \) and \( F_{2} \) are the left and right foci of \( C \), respectively. A line \( l \) passing through \( F_{2} \) intersects \( C \) at points \( A \) and \( B \) (with point \( A \) in the first quadrant), and \(\overrightarrow{A F_{2}} = 3 \overrightarrow{F_{2} B} \). Given that the area of triangle \( \triangle F_{1} A B \) is \( \frac{32}{3} \), find the radius of the incircle of triangle \( \triangle F_{1} A B \).

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