IMAC 2013 Problem 3

Let $ABC$ be a triangle with $\angle ABC=120^o$ and triangle bisectors $(AA_1),(BB_1),(CC_1)$ , respectively. $B_1F \perp A_1C_1$ , where $F\in (A_1C_1)$ . Let $R,I$ and $S$ be the centers of the circles which are inscribed in triangles $C_1B_1F,C_1B_1A_1, A_1B_1F$ , and $B_1S\cap A_1C_1=\{Q\}$ . Show that $R,I,S,Q$ are on the same circle.