<|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
 Let \( A_{1} A_{2} \ldots A_{k} \) be a regular \( k \)-gon inscribed in a circle of radius 1, and let \( P \) be a point lying on or inside the circumcircle. Find the maximum possible value of \( \left(P A_{1}\right)\left(P A_{2}\right) \cdots\left(P A_{k}\right) \). 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
2
Let \( A_{1} A_{2} \ldots A_{k} \) be a regular \( k \)-gon inscribed in a circle of radius 1, and let \( P \) be a point lying on or inside the circumcircle. Find the maximum possible value of \( \left(P A_{1}\right)\left(P A_{2}\right) \cdots\left(P A_{k}\right) \).
test