BxMO 2013 Problem 3

Let $\triangle A B C$ be a triangle with circumcircle $\Gamma$, and let $I$ be the center of the incircle of $\triangle A B C$. The lines $A I, B I$ and $C I$ intersect $\Gamma$ in $D \neq A, E \neq B$ and $F \neq C$. The tangent lines to $\Gamma$ in $F, D$ and $E$ intersect the lines $A I, B I$ and $C I$ in $R, S$ and $T$, respectively. Prove that

$$|A R| \cdot|B S| \cdot|C T|=|I D| \cdot|I E| \cdot|I F|$$