SHARYGIN 2011 Problem 11

The excircle of right-angled triangle $ABC$ ( $\angle B =90^o$ ) touches side $BC$ at point $A_1$ and touches line $AC$ in point $A_2$ . Line $A_1A_2$ meets the incircle of $ABC$ for the first time at point $A'$ , point $C'$ is defined similarly. Prove that $AC||A'C'$ .