
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
The side length of a regular $n$-gon is $a$, the radius of its circumscribed circle is $R$, and the radius of its inscribed circle is $r$. We draw a circle $k$ with radius $\varrho$ around its center such that

$$
R-\frac{a}{2} \leq \varrho \leq r
$$

Furthermore, around each vertex of the polygon, we draw circles that touch $k$ externally. The combined area covered by the circle $k$ and the $n$ touching circles partially covers the polygon. Construct the circle $k$ such that the area of the covered region is a) as large as possible; b) as small as possible.

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