TSTST 2023 Problem 1

Let $A B C$ be a triangle with centroid $G$. Points $R$ and $S$ are chosen on rays $G B$ and $G C$, respectively, such that $$\angle A B S=\angle A C R=180^{\circ}-\angle B G C .$$ Prove that $\angle R A S+\angle B A C=\angle B G C$.