RU-SPB 2020 Problem 105

Rays $\ell, \ell_1, \ell_2$ have the same starting point $O$ , such that the angle between $\ell$ and $\ell_2$ is acute and the ray $\ell_1$ lies inside this angle. The ray $\ell$ contains a fixed point of $F$ and an arbitrary point $L$ . Circles passing through $F$ and $L$ and tangent to $\ell_1$ at $L_1$ , and passing through $F$ and $L$ and tangent to $\ell_2$ at $L_2$ . Prove that the circumcircle of $\triangle FL_1L_2$ passes through a fixed point other than $F$ independent on $L$ .