A plane contains points $A$ and $B$ with $AB = 1$.  Let $S$ be the union of all disks of radius 1 in the plane that cover $\overline{AB}$. What is the area of $S$? Express your answer in terms of $\pi$ and in simplest radical form.
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
A plane contains points $A$ and $B$ with $AB = 1$.  Let $S$ be the union of all disks of radius 1 in the plane that cover $\overline{AB}$. What is the area of $S$? Express your answer in terms of $\pi$ and in simplest radical form.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
3\pi-\frac{\sqrt{3}}{2}
A plane contains points $A$ and $B$ with $AB = 1$.  Let $S$ be the union of all disks of radius 1 in the plane that cover $\overline{AB}$. What is the area of $S$? Express your answer in terms of $\pi$ and in simplest radical form.
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