A problem by Sagnik Saha

$\triangle ABC$ is an acute-angled triangle with $AB=13$ , $BC=14$ and $AC=15$ . The incircle $\Gamma$ of $ABC$ with centre $I$ touches $BC$ at $D$ . Let $AI$ produced meet $BC$ at $E$ . Let $K$ be the mid-point of $BC$ . $KI$ meets $AD$ at $L$ . Find the numerical value of

$\dfrac{[R_1] \times [R_2] \times [R_3]}{[ABC]-\frac{4}{3}[LDK]}.$

Details and assumptions

$R_1$ is the circumradius of $\triangle ABC$ , $R_2$ is the circumradius of $\triangle ALI$ , $R_3$ is the circumradius of $\triangle IDE$

$[PQRS]$ denotes the area of the figure $PQRS$ .