Let $\omega$ be a circle with center $O$, and let $l$ be a line not intersecting $\omega$. $E$ is a point on $l$ such that $OE$ is perpendicular with $l$. Let $M$ be an arbitrary point on $M$ different from $E$. Let $A$ and $B$ be distinct points on the circle $\omega$ such that $MA$ and $MB$ are tangents to $\omega$. Let $C$ and $D$ be the foot of perpendiculars from $E$ to $MA$ and $MB$ respectively. Let $F$ be the intersection of $CD$ and $OE$. As $M$ moves, determine the locus of $F$.