
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 that \( P \) is a point on the hyperbola \( C: \frac{x^{2}}{4} - \frac{y^{2}}{12} = 1 \), and \( F_{1} \) and \( F_{2} \) are the left and right foci of the hyperbola \( C \), and \( M \) and \( I \) are the centroid and incenter of the triangle \( \triangle P F_{1} F_{2} \) respectively. If \( M I \perp x \)-axis, then find the radius of the incircle of \( \triangle P F_{1} F_{2} \).

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