HMMT 2006 A7

Let

$$f(x)=x^{4}-6 x^{3}+26 x^{2}-46 x+65 .$$

Let the roots of $f(x)$ be $a_{k}+i b_{k}$ for $k=1,2,3,4$. Given that the $a_{k}, b_{k}$ are all integers, find $\left|b_{1}\right|+\left|b_{2}\right|+\left|b_{3}\right|+\left|b_{4}\right|$.