
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
An ellipse with the equation \(\frac{x^2}{9} + \frac{y^2}{4} = 1\) has one of its focal points and exactly three points where a circle is tangent to the ellipse. If the foci of the ellipse are at points \((\pm c, 0)\), find the radius \(r\) of such a circle that it is smallest.
A) \(3\)
B) \(1+\sqrt{5}\)
C) \(3+\sqrt{5}\)
D) \(3-\sqrt{5}\)
E) \(4-\sqrt{5}\)

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