What is the product of $0.\overline{05}$ and $1.8$ expressed as a fraction in simplest form?
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
What is the product of $0.\overline{05}$ and $1.8$ expressed as a fraction in simplest form?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
\frac{1}{11}
What is the product of $0.\overline{05}$ and $1.8$ expressed as a fraction in simplest form?
train