
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 sphere of radius 1, it contains equal circles \(\gamma_{0}, \gamma_{1}, \ldots, \gamma_{n}\) each of radius \(r\) (\(n \geq 3\)). Circle \(\gamma_{0}\) is tangent to all circles \(\gamma_{1}, \ldots, \gamma_{n}\). Moreover, \(\gamma_{1}\) and \(\gamma_{2}\), \(\gamma_{2}\) and \(\gamma_{3}\), ..., \(\gamma_{n}\) and \(\gamma_{1}\) are also tangent to each other. For which \(n\) is this possible? Compute the corresponding radius \(r\).

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