JOMO 5, Short 10

In the Diagram, $AB=10, BC=24$ . $I$ is the Incenter of $\triangle ABC$ , $G$ is the Centroid of $\triangle ABC$ . $O$ and $E$ are the midpoints of $AC$ and $BC$ respectively.

The product of $OG, BG, BI, IE, OE, GE, GK, KE$ and $IK$ can be expressed as $\frac{a\sqrt b}c$ where $a,b,c\in \mathbb Z^+$ , $gcd(a,c)=1$ , and $b$ is not divisible by the square of a integer larger than 1. Find the last 3 digits of $a+b+c$