HMIC 2023 Problem 3

Triangle $A B C$ has incircle $\omega$ and $A$-excircle $\omega_{A}$. Circle $\gamma_{B}$ passes through $B$ and is externally tangent to $\omega$ and $\omega_{A}$. Circle $\gamma_{C}$ passes through $C$ and is externally tangent to $\omega$ and $\omega_{A}$. If $\gamma_{B}$ intersects line $B C$ again at $D$, and $\gamma_{C}$ intersects line $B C$ again at $E$, prove that $B D=E C$.