KZ-MO 2016 Problem 3

Circles $\omega_1 , \omega_2$ intersect at points $X,Y$ and they are internally tangent to circle $\Omega$ at points $A,B$ ,respectively. $AB$ intersect with $\omega_1 , \omega_2$ at points $A_1,B_1$ ,respectively.Another circle is internally tangent to $\omega_1 , \omega_2$ and $A_1B_1$ at $Z$ .Prove that $\angle AXZ =\angle BXZ$ .(C.Ilyasov)

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