
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
21. $n$ wire triangles are arranged in space such that:

1. Any two of them share a common vertex.
2. The same number $k$ of triangles meet at each vertex.

Find all possible values of $k$ and $n$ for which this arrangement is possible.

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