
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 regular hexagon with side length \(a\) is rotated around one of the tangents to the circumscribed circle. What is the surface area described by the hexagon?

Also,
Given a circle rotated around a straight line in its plane which does not intersect it, resulting in a surface known as a torus.
Knowing the radius \(r\) of the rotating circle and the distance \(R\) from its center to the axis of rotation (\(R > r\)), find the surface area \(S\) of the torus.

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