Let \[f(x)=x^4-6x^3+26x^2-46x+65.\] Let the roots of $f(x)$ be $a_k+ib_k$ for $k=1,2,3,4$.  Given that the $a_k$, $b_k$ are all integers, find $|b_1|+|b_2|+|b_3|+|b_4|.$

Let's think step by step and output the final answer within \boxed{}.
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