Nemotron-IMO-Bench 2026 G11

Prove that for any inscribed hexagon $ABCDEF$ $$\frac{AC}{AB}\cdot \frac{BD}{BC}\cdot \frac{CE}{CD}\cdot \frac{DF}{DE}\cdot \frac{EA}{EF}\cdot \frac{FB}{FA}\geq 27.$$