<|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
 In the figure, point \(E\) is an arbitrary point on the diagonal \(AC\) of rectangle \(ABCD\). Lines are drawn through \(E\) perpendicular to \(AB\) and \(BC\), intersecting \(AB\) and \(BC\) at points \(F\) and \(G\) respectively. Segments \(DF\), \(FG\), and \(GD\) are then connected. Given that \(AB = 8\), \(AD = 10\), and the area of triangle \(DFG\) is 30, what is the area of rectangle \(BGEF\)? 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
In the figure, point \(E\) is an arbitrary point on the diagonal \(AC\) of rectangle \(ABCD\). Lines are drawn through \(E\) perpendicular to \(AB\) and \(BC\), intersecting \(AB\) and \(BC\) at points \(F\) and \(G\) respectively. Segments \(DF\), \(FG\), and \(GD\) are then connected. Given that \(AB = 8\), \(AD = 10\), and the area of triangle \(DFG\) is 30, what is the area of rectangle \(BGEF\)?
test