A circle of radius $1$ has diameter $\overline{AB}$ and chord $\overline{CD}$ perpendicular to $\overline{AB}$, intersecting $\overline{AB}$ at point $E$. If the maximum possible area of quadrilateral $ABCD$ is $M$, what is the smallest length of $AE$ so that the area of $ABCD = \frac{M}{n}$? Express your answer in terms of $n$.
Also, prove that for integer $n > 1$, length $AE$ is an irrational number.