Let $O$ be the center of a circle with radius $26$, and let $A$ and $B$ be two distinct points on the circle. Let $M$ be the midpoint of segment $AB$. Consider a point $C$ such that $CO = 34$ and $\angle COM = 15^\circ$. Let $N$ be the midpoint of segment $CO$. Suppose that $\angle ACB = 90^\circ$. Find the length of $MN$.

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