Given an acute triangle $ABC$. A circle inscribed in a triangle $ABC$ with center at point $I$ touches the sides $AB, BC$ at points $C_1$ and $A_1$, respectively. The lines $A_1C_1$ and $AC$ intersect at the point $Q$. Prove that the circles circumscribed around the triangles $AIC$ and $A_1CQ$ are tangent. (Dmitry Shvetsov)