Five. (20 points) Given the semi-ellipse $\frac{x^{2}}{4}+y^{2}=1(y>0)$, two perpendicular lines are drawn through a fixed point $C(1,0)$ intersecting the ellipse at points $P$ and $Q$, respectively. Here, $O$ is the origin, and $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse.
(1) Find the minimum value of $\left|P F_{1}+P F_{2}\right|$;
(2) Does there exist a line such that the sum of the x-coordinates of $P F_{1}$ and $P F_{2}$ is minimized? If not, explain why.