<|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 convex quadrilateral \(ABCD\), the midpoints of sides \(BC\) and \(CD\) are \(E\) and \(F\) respectively. The segments \(AE\), \(EF\), and \(AF\) divide the quadrilateral into four triangles whose areas are four consecutive integers. What is the maximum possible area of triangle \(ABD\)? 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
6
In the convex quadrilateral \(ABCD\), the midpoints of sides \(BC\) and \(CD\) are \(E\) and \(F\) respectively. The segments \(AE\), \(EF\), and \(AF\) divide the quadrilateral into four triangles whose areas are four consecutive integers. What is the maximum possible area of triangle \(ABD\)?
test