JOMO 5, Short 10

Geometry Level 5

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 \(a+b+c\)

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