The chord $[CD]$ of the circle with diameter $[AB]$ is perpendicular to $[AB]$. Let $M$ and $N$ be the midpoints of $[BC]$ and $[AD]$, respectively. If $|BC|=6$ and $|AD|=2\sqrt{3}$, find the length of $|MN|$. The original answer is in the form \(\sqrt{m}\). Please find the value of \(m\).

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