13. Given a positive number $r$ such that the set $T=\left\{(x, y) \mid x, y \in \mathbf{R}\right.$, and $\left.x^{2}+(y-7)^{2} \leqslant r^{2}\right\}$ is a subset of the set $S=\{(x, y) \mid x$, $y \in \mathbf{R}$, and for any $\theta \in \mathbf{R}$, $\cos 2 \theta+x \cos \theta+y \geqslant 0\}$, find the maximum value of $r$.
13. Set $S$ represents the part of the plane above the line $l: x \cos \theta+y+\cos 2 \theta=0$ (including the line $l$ itself), and set $T$ represents the circle $x^{2}+(y-7)^{2}=r^{2}$ and its interior. Since $T \subseteq S$, the distance $d$ from the center $(0,7)$ to the line $l$ is $d=\frac{7+\cos 2 \theta}{\sqrt{1+\cos ^{2} \theta}}$ $=\frac{4}{\sqrt{1+\cos ^{2} \theta}}+2 \sqrt{1+\cos ^{2} \theta} \geqslant 4 \sqrt{2}$, with equality when $\cos \theta= \pm 1$. Therefore, the maximum value of $r$ is $4 \sqrt{2}$.