55th IMO Team Selection Test

Given is a $△ABC$ with incircle $k$ touching the sides $BC$ and $CA$ in points $P$ and $Q$ respectively. Let denote with $J$ the center of the excircle at side $AB$ in $\angle ABC$ and with $T$ – the second intersection point of the circumcircles of $\angle JBP$ and $\angle JAQ$. Prove that the circumcircle of $\angle ABT$ touches $k$.