OMCC 2018 Problem 2

Let $\Delta ABC$ be a triangle inscribed in the circumference $\omega$ of center $O$ . Let $T$ be the symmetric of $C$ respect to $O$ and $T'$ be the reflection of $T$ respect to line $AB$ . Line $BT'$ intersects $\omega$ again at $R$ . The perpendicular to $CT$ through $O$ intersects line $AC$ at $L$ . Let $N$ be the intersection of lines $TR$ and $AC$ . Prove that $\overline{CN}=2\overline{AL}$ .