Nemotron-IMO-Bench 2026 G45

In an acute-angled triangle $A_0A_1A_2$ with circumcenter $0$ the lines $OA_k$ and $A_{k+1}A_{k+2}$ meet at $B_k$ for $k=0,1,2$ (the indices are taken modulo $3$). The tangent to the circumcircle of $A_0A_1A_2$ at $A_k$ meets the line $B_{k+1}B_{k+2}$ at $C_k$. Prove that $C_0,C_1,C_2$ are collinear.