<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 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. Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
1024
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.
test