SHARYGIN 2023 Problem 88

Two circles $\omega_1$ and $\omega_2$ meeting at point $A$ and a line $a$ are given. Let $BC$ be an arbitrary chord of $\omega_2$ parallel to $a$ , and $E$ , $F$ be the second common points of $AB$ and $AC$ respectively with $\omega_1$ . Find the locus of common points of lines $BC$ and $EF$ .