Square $ABCD$ has side length $1$; circle $\Gamma$ is centered at $A$ with radius $1$. Let $M$ be the midpoint of $BC$, and let $N$ be the point on segment $CD$ such that $MN$ is tangent to $\Gamma$. Compute $MN$.

[i]2018 CCA Math Bonanza Individual Round #11[/i]