
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
The $\mathcal{K}$ is a right circular cone with an inscribed sphere $\mathcal{G}$. The centers of the spheres $g_{1}, g_{2}, \ldots, g_{n}$, each with radius $r$, form a regular $n$-gon with side length $2r$. Additionally, each sphere $g_{i}$ touches both the lateral surface and the base of $\mathcal{K}$, as well as the sphere $\mathcal{G}$. What values can $n$ take?

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