<|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
 The real numbers \( x_1, x_2, \ldots, x_{1997} \) have a sum of \(-318\sqrt{3}\) and satisfy \(-\frac{1}{\sqrt{3}} \le x_i \le \sqrt{3}\). What is the maximum possible value for the sum of their 12th powers? 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
189548
The real numbers \( x_1, x_2, \ldots, x_{1997} \) have a sum of \(-318\sqrt{3}\) and satisfy \(-\frac{1}{\sqrt{3}} \le x_i \le \sqrt{3}\). What is the maximum possible value for the sum of their 12th powers?
test