solution: This is a geometric sequence with first term $\frac{1}{4}$ and common ratio $\frac{1}{2}$. Thus the sum of the first $n$ terms is:
  
  $\frac{255}{512}=\frac{1}{4}\left(\frac{1-\left(\frac{1}{2}\right)^n}{1-\frac{1}{2}}\right)=\frac{2^n-1}{2^{n+1}}$.
  
  We see that $\frac{255}{512}=\frac{2^8-1}{2^9}$, so $n=\boxed{8}$.
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  The sum of the first $n$ terms in the infinite geometric sequence $\left\{\frac{1}{4},\frac{1}{8},\frac{1}{16},\dots \right\}$ is $\frac{255}{512}$. Find $n$.
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: 8
extra_info.question: The sum of the first $n$ terms in the infinite geometric sequence $\left\{\frac{1}{4},\frac{1}{8},\frac{1}{16},\dots \right\}$ is $\frac{255}{512}$. Find $n$.
extra_info.split: test