KöMaL 2020 C1588

Let $E$ and $F$ be the points lying closer to vertex $A$ which divide the sides $AB$ and $AD$ of a quadrilateral $ABCD$, respectively, in a $1:2$ ratio. Let $G$ be the point lying closer to $B$ which divides side $BC$ in a $1:2$ ratio. Reflect point $G$ in the point $E$, and reflect its image in the point $F$. Prove that the final image lies on a side of the quadrilateral. Which side is it, and in what ratio is it divided by the final image?