
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
Let  $S_1$  and  $S_2$  be two spheres with distinct radii that touch externally. The spheres lie inside a cone  $C$ , and each sphere touches the cone in a full circle. Inside the cone there are  $n$  additional solid spheres arranged in a ring in such a way that each solid sphere touches the cone  $C$ , both of the spheres  $S_1$  and  $S_2$  externally, as well as the two neighboring solid spheres. What are the possible values of  $n$ ?

*Proposed by Iceland.*

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