For convex cyclic hexagon $ABCDEF$, show $AC\cdot BD \cdot CE \cdot DF \cdot EA\cdot FB \geq 27\cdot AB\cdot BC\cdot CD \cdot DE\cdot EF\cdot FA$

Question

Given a convex hexagon $ABCDEF$ inscribed in the circle, prove that $$AC\cdot BD \cdot CE \cdot DF \cdot EA\cdot FB \;\geq\; 27\cdot AB\cdot BC\cdot CD \cdot DE\cdot EF\cdot FA$$

("$AC$" means the length of the segment with endpoints $A$ and $C$.)

What I did until now: bashing complex and making problem even harder!