
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 of radius \( r \) is concentric with and outside a regular hexagon with side length \( 3 \). Determine the radius \( r \) such that the probability that two entire sides of the hexagon are visible from a randomly chosen point on the circle is \( \frac{1}{4} \).
A) \( \frac{9\sqrt{3}}{\sqrt{3 + 3\sqrt{2}}} \)
B) \( \frac{6\sqrt{3}}{\sqrt{2 + \sqrt{2}}} \)
C) \( 3\sqrt{3} + \sqrt{2} \)
D) \( 6\sqrt{3} - \sqrt{2} \)

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