Problem 538

[ $\quad$ Tangents to Spheres $\quad$ ]

In a regular triangular pyramid $S A B C$ ( $S$ - the apex), the side of the base is 6, and the height of the pyramid $S H$ is $\sqrt{15}$. A plane passes through point $B$ perpendicular to line $A S$ and intersects segment $S H$ at point $O$. Points $P$ and $Q$ are located on lines $A S$ and $C B$ respectively, such that line $P Q$ is tangent to a sphere of radius $\sqrt{\frac{2}{5}}$ with center at point $O$. Find the minimum length of segment $P Q$.