Silk Road Mathematics Competition

In triangle $ABC$ the angle bisectors of $A$ and $C$ intersect the sides $BC$ and $AB$ at the points $A_1$ and $C_1$, respectively, and the circumcircle of the triangle $ABC$ at the points $A_2$ and $C_2$, respectively. Let $K$ be the point of intersection of $A_1{C}_2$ and $C_1{A}_2$, and $I$ be the incenter of triangle $ABC$. Prove that $KI$ passes through the midpoint of $AC$.