BMO Round 1, 2022

Two circles $\Gamma_1$ and $\Gamma_2$ have centres $O_1$ and $O_2$ respectively. They pass through each other's centres and intersect at $A$ and $B$. The point $C$ lies on the minor arc $BO_2$ of $\Gamma_1$. The points $D$ and $E$ lie on the line $O_2C$ such that $\angle AO_1D = \angle DO_1C$ and $\angle CO_1E = \angle EO_1B$. Prove that triangle $DO_1E$ is equilateral.

(A minor arc of a circle is the shorter of the two arcs with given endpoints.)