<|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
 An ellipse satisfies the following optical property: a ray of light emitted from one focus of the ellipse, after being reflected by the ellipse, passes through the other focus. Now, consider a horizontally placed elliptical billiard table satisfying the equation: $\dfrac{x^2}{16} + \dfrac{y^2}{9} = 1$, where points $A$ and $B$ are its two foci. When a stationary ball is placed at point $A$ and starts moving in a straight line, after being reflected by the elliptical wall (not at the endpoints of the major axis) and returning to point $A$, what is the shortest distance the ball travels? Express your answer as a single number. 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
16
An ellipse satisfies the following optical property: a ray of light emitted from one focus of the ellipse, after being reflected by the ellipse, passes through the other focus. Now, consider a horizontally placed elliptical billiard table satisfying the equation: $\dfrac{x^2}{16} + \dfrac{y^2}{9} = 1$, where points $A$ and $B$ are its two foci. When a stationary ball is placed at point $A$ and starts moving in a straight line, after being reflected by the elliptical wall (not at the endpoints of the major axis) and returning to point $A$, what is the shortest distance the ball travels? Express your answer as a single number.
test