7.2. In triangle $K L M$, the median $K P$ is drawn, point $O$ is the center of the circumscribed circle around it, and point $Q$ is the center of the inscribed circle in it. Segments $K P$ and $O Q$ intersect at point $S$, and $\frac{O S}{P S}=\sqrt{6} \frac{Q S}{K S}$. Find the product of the cosines of the angles $K L M$ and $K M L$, given that $\angle L K M=\frac{\pi}{3}$. Round the answer to the nearest hundredth if necessary.
Answer: $-0.38$ (exact value: $-3 / 8$ ).