<|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 triangle \( \triangle ABC \), \( D \) is the midpoint of side \( BC \). Given that \( \overrightarrow{AD} \cdot \overrightarrow{AC} = 0 \), calculate the value of \( \tan A + 2 \tan C \). Express your answer as a single numerical value. 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
0
In the triangle \( \triangle ABC \), \( D \) is the midpoint of side \( BC \). Given that \( \overrightarrow{AD} \cdot \overrightarrow{AC} = 0 \), calculate the value of \( \tan A + 2 \tan C \). Express your answer as a single numerical value.
test