JBMO-SL 2004 G5

Let $A B C$ be a triangle with $\angle C=90^{\circ}$ and $D \in C .4, E \in C^{\prime} B$, and $k_{1}, k_{2}, k_{3}, k_{4}$ semicircles with diameters $C A, C B, C D, C E$ respectively, which have common part with the triangle $A B C$. Let also,

$$k_{1} \cap k_{2}=\{C, K\}, k_{3} \cap k_{4}=\{C, M\}, k_{2} \cap k_{3}=\{C, L\}, k_{1} \cap k_{4}=\{C, N\}$$

Prove that $K, L, M$ and $N$ are cocyclic points.