
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 $n$-pointed star to be the union of $n$ line segments $Q_1Q_2, Q_2Q_3, \ldots, Q_nQ_1$ such that:
- The points $Q_1, Q_2, \ldots, Q_n$ are coplanar, and no three of them are collinear.
- Each of the $n$ line segments intersects at least one of the other line segments at a point other than an endpoint.
- All of the angles at $Q_1, Q_2, \ldots, Q_n$ are congruent.
- All of the $n$ line segments are congruent.
- The path $Q_1Q_2, Q_2Q_3, \ldots, Q_nQ_1$ turns counterclockwise at an angle of less than 180 degrees at each vertex.
- A segment from $Q_i$ to $Q_{i+2}$ is considered (skipping one vertex in between).
Determine how many non-similar regular $12$-pointed stars are there.

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