Suppose that $a, b, c$ are positive reals such that
$\begin{aligned}
& x^3+7x^2+24x+18 \\
&= (x+a)(x+b+ci)(x+b-ci),
\end{aligned}$
where $i$ is the imaginary number that satisfies $i^2=-1.$ What is the value of $a+b+c?$

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