DE-MO 2013 Problem 3

Given two circles $k_1$ and $k_2$ which intersect at $Q$ and $Q'.$ Let $P$ be a point on $k_2$ and inside of $k_1$ such that the line $PQ$ intersects $k_1$ in a point $X\ne Q$ and such that the tangent to $k_1$ at $X$ intersects $k_2$ in points $A$ and $B.$ Let $k$ be the circle through $A,B$ which is tangent to the line through $P$ parallel to $AB.$ Prove that the circles $k_1$ and $k$ are tangent.