Christmas Contest Individual Round Tiebreaker

This was an improvised tie breaker problem to decide the winner in the individual round.

Let $\triangle ABC$ be acute-angled with $\angle ABC = 35^\circ$. Suppose that $I$ is the incenter of $\triangle ABC$ and that $AI +AC = BC$. Let $P$ be the point such that $PB$ and $PC$ are tangent to the circumcircle of $\triangle ABC$. Let $Q$ be the point on the line $AB$ such that $PQ$ is parallel to $AC$. Find $\angle AQC$ in degrees.