
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 F is the left focus of the hyperbola $C: \frac{x^2}{4} - \frac{y^2}{12} = 1$, point A is (1,4), and P is a point on the right branch of C. When the perimeter of triangle APF is minimized, the distance from point F to the line AP is ______.

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