Regular dodecahedron \(ABCDEFGHIJKL\) with side length \(1\) is drawn. Segments \(AF\), \(BG\), and \(CH\) are then drawn.

The area of the tiny little triangle formed from the intersections of these segments can be expressed as \[\dfrac{\sqrt{a}-b}{c}\] for positive integers \(a,b,c\). Find the value of \(a+b+c\).

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