Question 119
Let $ABC$ be an acute triangle with $AB \lt AC$ and $AB \lt BC$. Let $P$ be a point on the segment $BC$ such that $\angle APB = \angle BAC$. The tangent to the circumcircle of $\triangle ABC$ at $A$ meets the circumcircle of $\triangle APB$ at a point $Q \neq A$. Let $Q'$ be the reflection of $Q$ relative to the midpoint of $AB$. If line $PQ$ meets the segment $AQ'$ at $S$, prove that
$$
\frac{1}{AB} + \frac{1}{AC} \gt \frac{1}{CS}.
$$
$$
\frac{1}{AB} + \frac{1}{AC} \gt \frac{1}{CS}.
$$