IZhO 2019 Problem 4

An isosceles triangle $A B C$ with $A C=B C$ is given. Point $D$ is chosen on the side $A C$. The circle $S_{1}$ of radius $R$ with the center $O_{1}$ touches the segment $A D$ and the extensions of $B A$ and $B D$ over the points $A$ and $D$, respectively. The circle $S_{2}$ of radius $2 R$ with the center $O_{2}$ touches the segment $D C$ and the extensions of $B D$ and $B C$ over the points $D$ and $C$, respectively. Let the tangent to the circumcircle of the triangle $B O_{1} O_{2}$ at the point $O_{2}$ intersect the line $B A$ at point $F$. Prove that $O_{1} F=O_{1} O_{2}$.