
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
A circle passes through one focus of an ellipse, and touches the ellipse tangentially at one point. The ellipse is given by the equation \(x^2 + 4y^2 = 16\). Find the radius \(r\) of the circle.
A) \(2\sqrt{3}\)
B) \(4\)
C) \(4 - 2\sqrt{3}\)
D) \(2\sqrt{12}\)
E) \(5 - 2\sqrt{3}\)

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