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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 $A$ .