10. (20 points) As shown in Figure 2, the ellipse $C: \frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ has left and right vertices $A$ and $B$, respectively. Take any point $P$ on the ellipse $C$ different from $A$ and $B$. The lines $PA$ and $PB$ intersect the line $x=3$ at points $M$ and $N$, respectively. The line $MB$ intersects the ellipse $C$ at point $Q$.
(1) Find the value of $\overrightarrow{F M} \cdot \overrightarrow{F N}$;
(2) Prove that points $A$, $Q$, and $N$ are collinear.