TR-MO 1999 Problem 2

Problem-2: Given a circle with center $O$ , the two tangent lines from a point $S$ outside the circle touch the circle at points $P$ and $Q$ . Line $SO$ intersects the circle at $A$ and $B$ , with $B$ closer to $S$ . Let $X$ be an interior point of minor arc $PB$ , and let line $OS$ intersect lines $QX$ and $PX$ at $C$ and $D$ , respectively. Prove that $\frac{1}{\left| AC \right|}+\frac{1}{\left| AD \right|}=\frac{2}{\left| AB \right|}$ .