Problem 23 (2018 AMC 12A)

In $\triangle ABC$, $AB=6$, $BC=7$, and $CA=8$. Point $D$ lies on $\overline{BC}$, and $\overline{AD}$ bisects $\angle BAC$. Point $E$ lies on $\overline{AC}$, and $\overline{BE}$ bisects $\angle ABC$. The bisectors $\overline{AD}$ and $\overline{BE}$ intersect at $I$. What is the ratio $AI : ID$?