SELECTION EXAMINATION

Let $ABC$ be a scalene acute angled triangle with $AB<AC<BC$ and circumcenter $c(O,R)$. The ex-circle ($c_A$) corresponding to the vertex $A$, has center $I$ and is tangent to the sides $BC$, $AC$, $AB$ at $D$, $E$, $Z$, respectively. The line $AI$ intersects the circle $c(O,R)$ at $M$ and the circumcircle, say $(c_1)$, of the triangle $AZE$ intersects the circle ($c$) at $K$. The circumcircle, say $(c_2)$, of the triangle $OKM$ intersects the circle ($c_1$) at point $N$. Prove that the lines $AN$ and $KI$ intersect at a point of the circle ($c$).