jmc

Plugging in, we have $w=\frac{3(1+i)+1}{5(1+i)+7}=\frac{4+3i}{12+5i}$. 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 $b$, $\left|\frac{a}{b}\right|=\frac{|a|}{|b|}$. 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 $|w|=\boxed{\frac{5}{13}}$.