<|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  $ ABCD$  be an isosceles trapezoid with  $ \overline{AD}\parallel{}\overline{BC}$  whose angle at the longer base  $ \overline{AD}$  is  $ \dfrac{\pi}{3}$ . The diagonals have length  $ 10\sqrt {21}$ , and point  $ E$  is at distances  $ 10\sqrt {7}$  and  $ 30\sqrt {7}$  from vertices  $ A$  and  $ D$ , respectively. Let  $ F$  be the foot of the altitude from  $ C$  to  $ \overline{AD}$ . The distance  $ EF$  can be expressed in the form  $ m\sqrt {n}$ , where  $ m$  and  $ n$  are positive integers and  $ n$  is not divisible by the square of any prime. Find  $ m \plus{} n$ . 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
32
Let  $ ABCD$  be an isosceles trapezoid with  $ \overline{AD}\parallel{}\overline{BC}$  whose angle at the longer base  $ \overline{AD}$  is  $ \dfrac{\pi}{3}$ . The diagonals have length  $ 10\sqrt {21}$ , and point  $ E$  is at distances  $ 10\sqrt {7}$  and  $ 30\sqrt {7}$  from vertices  $ A$  and  $ D$ , respectively. Let  $ F$  be the foot of the altitude from  $ C$  to  $ \overline{AD}$ . The distance  $ EF$  can be expressed in the form  $ m\sqrt {n}$ , where  $ m$  and  $ n$  are positive integers and  $ n$  is not divisible by the square of any prime. Find  $ m \plus{} n$ .
test