UZ-MO 2013 Problem 4

Let circles $\Gamma$ and $\omega$ are circumcircle and incircle of the triangle $ABC$ , the incircle touches sides $BC,CA,AB$ at the points $A_1,B_1,C_1$ . Let $A_2$ and $B_2$ lies the lines $A_1I$ and $B_1I$ ( $A_1$ and $A_2$ lies different sides from $I$ , $B_1$ and $B_2$ lies different sides from $I$ ) such that $IA_2=IB_2=R$ . Prove that : (a) $AA_2=BB_2=IO$ ; (b) The lines $AA_2$ and $BB_2$ intersect on the circle $\Gamma ;$