In \(\triangle FGH\), \(FG = 7\), \(GH = 8\), and \(HF = 9\) with \(FP\) as the perpendicular. \(HJ\) and \(GI\) are the angle bisectors of \(\angle FHG\) and \(\angle FGH\) respectively. The lines \(HJ\) and \(GI\) intersect the line \(FP\) at points \(R\) and \(S\) respectively. If the length of \(RS\) can be expressed in the form \(\frac{y}{z}\), what is the minimum value of \(y + z\)?

Geometry

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