BALKAN-SL 2015 G1

In an acute angled triangle $ABC$ , let $BB'$ and $CC'$ be the altitudes. Ray $C'B'$ intersects the circumcircle at $B''$ andl let $\alpha_A$ be the angle $\widehat{ABB''}$ . Similarly are defined the angles $\alpha_B$ and $\alpha_C$ . Prove that $$\sin \alpha _A \sin \alpha _B \sin \alpha _C\leq \frac{3\sqrt{6}}{32}$$ (Romania)