
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
Given a regular hexagon, draw a circle $k$ around its center with a radius no greater than the radius of the circle that is tangent to the sides of the hexagon. Then, draw a circle around each vertex of the hexagon that is tangent to $k$ and has a diameter no greater than the side length of the hexagon. The 7 circles together cover part of the hexagon. What is the radius of the circle $k$ if the covered area is minimal, and what is it if the covered area is maximal?

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