solution: We can split the expression $|5x-1|=x+3$ into two separate cases. In the first case, \begin{align*} 5x-1&=x+3
  \\\Rightarrow \qquad 4x&=4
  \\\Rightarrow \qquad x&=1
  \end{align*}If we plug this value of $x$ back into the original equation in order to check our answer, we see that $|5(1)-1|=1+3$ or $4=4$. Since this is true, we can accept $x=1$ as a valid solution.
  
  In the second case, \begin{align*} 5x-1&=-(x+3)
  \\ 5x-1&=-x-3
  \\\Rightarrow \qquad 6x&=-2
  \\\Rightarrow \qquad x&=-\frac13.
  \end{align*}If we plug $-\frac13$ back into the initial equation, we get that $\left|5\left(-\frac13\right)-1\right|=-\frac13+3$, which simplifies to $\left|-\frac{8}{3}\right|=\frac{8}{3}$, or $\frac{8}{3}=\frac{8}{3}$. Since this is also a true statement, we can accept $x=-\frac13$ as a valid solution as well. Since both $x=1$ and $x=-\frac13$ are possible values of $x$, the final answer is the larger of the two: $x=\boxed{1}$.
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  Find the largest value of $x$ that satisfies the equation $|5x-1|=x+3$.
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: 1
extra_info.question: Find the largest value of $x$ that satisfies the equation $|5x-1|=x+3$.
extra_info.split: test