Consider a triangle $ABC$ where $AB = AC = 15$ and $BC = 18$. Points $D$ and $E$ are chosen on $CA$ and $CB$, respectively, such that $CD = 5$ and $CE = 3$. The point $F$ is chosen on the half-line $\overrightarrow{DE}$ so that $EF = 8$. If $M$ is the midpoint of $AB$ and $N$ is the intersection of $FM$ and $BC$, what is the length of $CN$?

Let's think step by step and output the final answer within \boxed{}.
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