MathNet 2013 Question 4

Consider triangle $ABC$ with $\angle A = 2 \angle B$. The angle bisectors from $A$ and $C$ intersect at $D$, and the angle bisector from $C$ intersects $\overline{AB}$ at $E$. If $\frac{DE}{DC} = \frac{1}{3}$, compute $\frac{AB}{AC}$.

Supplemental illustration (AI-generated):