AMC 12 2018b Problem 12

Side $\overline{AB}$ of $\triangle ABC$ has length $10$. The bisector of angle $A$ meets $\overline{BC}$ at $D$, and $CD = 3$. The set of all possible values of $AC$ is an open interval $(m,n)$. What is $m+n$?

$\textbf{(A) }16 \textbf{(B) }17 \textbf{(C) }18 \textbf{(D) }19 \textbf{(E) }20$