# Are you in or are you out

**Geometry**Level 5

Let \(ABCP\) be a quadrilateral inscribed in a circle \(\Gamma\) such that \(PB=PC\) and \(\angle BAC \) is obtuse. Let \(I\) be the incenter of triangle \(ABC\) and suppose that line \(PI\) intersects again \(\Gamma\) at point \(J\) (\(J\) belongs to the major arc \(BC\)). If \(BJ=10\), \(JC=17\) and \(\sin \angle BJC=\frac{77}{85}\), then the value of \(\frac{AB}{AC}\) can be written as \( \frac{a}{b} \), where \(a\) and \(b\) are coprime positive integers. What is the value of \(a+b\)?

**Details and assumptions**

The order of the vertices is \(A, B, C, P \). In particular, \(P\) lies on the minor arc of \(BC\) which contains \(A\).

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