SHARYGIN 2016 Problem 13

Given are a triangle $ABC$ and a line $\ell$ meeting $BC, AC, AB$ at points $L_a, L_b, L_c$ respectively. The perpendicular from $L_a$ to $BC$ meets $AB$ and $AC$ at points $A_B$ and $A_C$ respectively. Point $O_a$ is the circumcenter of triangle $AA_bA_c$. Points $O_b$ and $O_c$ are defined similarly. Prove that $O_a, O_b$ and $O_c$ are collinear.