
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
Points \( A_{1}, \ldots, A_{12} \) are the vertices of a regular 12-gon. How many different 11-segment open non-self-intersecting polygonal paths with vertices at these points exist? Paths that can be transformed into each other through rotation are considered the same.

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