HMMT 2020 G4

Let $A B C D$ be a rectangle and $E$ be a point on segment $A D$. We are given that quadrilateral $B C D E$ has an inscribed circle $\omega_{1}$ that is tangent to $B E$ at $T$. If the incircle $\omega_{2}$ of $A B E$ is also tangent to $B E$ at $T$, then find the ratio of the radius of $\omega_{1}$ to the radius of $\omega_{2}$.