
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
How many circles of radius 1 can be placed tangent to a given unit circle \( K \) such that:

a) No two of these circles intersect;

b) None of the circles contains the center of another circle;

c) How many circles of radius 1 can be arranged on a plane such that they all intersect the given unit circle \( K \), but none of them contains the center of \( K \) or the center of any other circle under consideration?

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