In triangle $ABC,$ $\angle A = 90^\circ,$ $AC = 1,$ and $AB = 5.$  Point $D$ lies on ray $\overrightarrow{AC}$ such that $\angle DBC = 2 \angle CBA.$  Compute $AD.$
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
In triangle $ABC,$ $\angle A = 90^\circ,$ $AC = 1,$ and $AB = 5.$  Point $D$ lies on ray $\overrightarrow{AC}$ such that $\angle DBC = 2 \angle CBA.$  Compute $AD.$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
\frac{37}{11}
In triangle $ABC,$ $\angle A = 90^\circ,$ $AC = 1,$ and $AB = 5.$  Point $D$ lies on ray $\overrightarrow{AC}$ such that $\angle DBC = 2 \angle CBA.$  Compute $AD.$
train