4. Given the set $M=\left\{(x, y) \left\lvert\, y \geqslant \frac{1}{4} x^{2}\right.\right\}$,
$$
\begin{array}{l}
N=\left\{(x, y) \left\lvert\, y \leqslant-\frac{1}{4} x^{2}+x+7\right.\right\}, \\
D_{r}\left(x_{0}, y_{0}\right)=\left\{(x, y)\left|\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2} \leqslant r^{2}\right| .\right.
\end{array}
$$

Find the largest $r$ such that $D_{r}\left(x_{0}, y_{0}\right) \subset M \cap N$.
(1st Chern Shiing-Shen Cup National High School Mathematics Olympiad)
Establish a Cartesian coordinate system, then $M \cap N$ represents a figure symmetric about the point $(1,4)$. Furthermore, $D_{r}\left(x_{0}, y_{0}\right)$ is the largest inscribed circle of the figure $M \cap N$. Therefore,
$$
r_{\max }=\sqrt{\frac{25-5 \sqrt{5}}{2}} .
$$