
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
Define a regular 12-pointed double-layered star where each vertex connects to two others, skipping some vertices between connections, and no three points of connections are collinear. Vertices are labeled $1, 2, \ldots, 12$. For each vertex like $P_k$, connect it to $P_{k+m \mod 12}$ and $P_{k+2m \mod 12}$ where $m$ is chosen such that $\gcd(m, 12) = 1$. Determine how many non-similar regular 12-pointed double-layered stars can be made.

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