solution: Plugging in, we have $w = \dfrac{3(1+i)+1}{5(1+i)+7} = \dfrac{4+3i}{12+5i}$. We could write this in the form $a+bi$ and take the magnitude, but it's easier to use the fact that, for all complex numbers $a$ and $b$, $\left|\dfrac{a}{b}\right| = \dfrac{|a|}{|b|}$. The magnitude of the numerator is $\sqrt{3^2+4^2} = \sqrt{25} = 5$, and the magnitude of the denominator is $\sqrt{12^2 + 5^2} = \sqrt{169} = 13$. So $|w| = \boxed{\frac{5}{13}}$.
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  Let $z = 1+i$ and $w = \dfrac{3z+1}{5z+7}$. Find $|w|$.
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: \frac{5}{13}
extra_info.question: Let $z = 1+i$ and $w = \dfrac{3z+1}{5z+7}$. Find $|w|$.
extra_info.split: test