Kick that altitude

Geometry Level 4

\(ABC\) is an acute triangle with an area of \(18\). Let \(P\) be the foot of the perpendicular from \(A\) to \(BC\) and let \(Q\) be the foot of the perpendicular from \( C \) to \(AB\). The area of the triangle \(BPQ\) is \(2,\) and the length of \(PQ\) is \(2\sqrt{2}.\)

Let \( \Gamma\) be the circumcircle of \(ABC\). The radius of \(\Gamma\) can be written as \(\frac{a}{b},\) where \(a\) and \(b\) are coprime positive integers. Find \(a+b.\)

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