RU-ALLRUSSIAN 2021 Problem 114

In triangle $ABC$ angle bisectors $AA_{1}$ and $CC_{1}$ intersect at $I$ . Line through $B$ parallel to $AC$ intersects rays $AA_{1}$ and $CC_{1}$ at points $A_{2}$ and $C_{2}$ respectively. Let $O_{a}$ and $O_{c}$ be the circumcenters of triangles $AC_{1}C_{2}$ and $CA_{1}A_{2}$ respectively. Prove that $\angle{O_{a}BO_{c}} = \angle{AIC}$