BALTIC-WAY 1999 Problem 15

Let $A B C$ be a triangle with $\angle C=60^{\circ}$ and $|A C|<|B C|$. The point $D$ lies on the side $B C$ and satisfies $|B D|=|A C|$. The side $A C$ is extended to the point $E$ where $|A C|=|C E|$. Prove that $|A B|=|D E|$.