Four. (15 points) As shown in Figure 3, in the Cartesian coordinate system $x O y$, $\odot O: x^{2}+y^{2}=4$ intersects the positive half-axis of the $x$-axis at point $A$, $\odot A:(x-2)^{2}+y^{2}=r^{2}(r>0)$ intersects $\odot O$ at points $B$ and $C$.
(1) Find the minimum value of $\overrightarrow{A B} \cdot \overrightarrow{A C}$;
(2) Let $P$ be any point on $\odot O$ other than $B$ and $C$, and let the lines $P B$ and $P C$ intersect the $x$-axis at points $M$ and $N$, respectively. Find the maximum value of $S_{\triangle P O M} S_{\triangle P O N}$.