
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
Provide an analysis to determine how many non-similar regular 12-pointed stars can be designed following these rules:
1. The points $P_1, P_2, \ldots, P_{12}$ are coplanar and no three of them are collinear.
2. Each segment $P_1P_2, P_2P_3,\ldots, P_{12}P_1$ intersects at least one other segment non-trivially (not at the endpoints).
3. All angles at points $P_1, P_2, \ldots, P_{12}$ are congruent.
4. All line segments $P_1P_2, \ldots, P_{12}P_1$ are congruent.
5. The path turns counterclockwise with an angle less than 180 degrees at each vertex but skips at least one vertex between connections.

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