SHARYGIN 2010 Problem 5

A point $E$ lies on the altitude $BD$ of a triangle $ABC$, and $\angle AEC = 90^\circ$. Points $O_1$ and $O_2$ are the circumcenters of the triangles $AEB$ and $CEB$; points $F, L$ are the midpoints of the segments $AC$ and $O_1O_2$. Prove that the points $L, E, F$ are collinear.