<|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 area of the closed figure enclosed by the parabola \( y^{2} = a x \) (with \( a > 0 \)) and the line \( x = 1 \) is \( \frac{4}{3} \). What is the coefficient of the term containing \( x^{-18} \) in the expansion of the binomial \( \left( x + \frac{a}{x} \right)^{20} \)? 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
20
The area of the closed figure enclosed by the parabola \( y^{2} = a x \) (with \( a > 0 \)) and the line \( x = 1 \) is \( \frac{4}{3} \). What is the coefficient of the term containing \( x^{-18} \) in the expansion of the binomial \( \left( x + \frac{a}{x} \right)^{20} \)?
test