MathNet 2014 Question 7

In triangle $ABC$, $I$ is the incentre. A circle is tangent to $AI$ at $I$ and passes through $B$. It meets $AB$ again at $P$ and $BC$ again at $Q$. Line $QI$ meets $AC$ at $R$. Prove that $|AR|\cdot|BQ|=|PI|^2$.