
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 circle $\odot(O, r)$ with radius $r$ that contains a set $N$ of $n$ points, where the minimum distance between any two points in $N$ is $a$. Find the maximum value of $n$ when $a$ is (1) $\sqrt{3} r$, (2) $\sqrt{2} r$, and (3) $r$.

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