10. Given the standard equation of ellipse $C$ as $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>1)$, point $P$ is a moving point on the unit circle, points $A, F$ are the right vertex and right focus of ellipse $C$ respectively, and when $\angle P O A=\frac{\pi}{3}$, $|P F|=1,|P A|=\sqrt{3}$.
(1) Find the standard equation of ellipse $C$;
(2) Draw a line $l$ through point $P$ intersecting ellipse $C$ at points $M, N$, and satisfying $P$ as the midpoint of segment $M N$. Then draw the tangents to ellipse $C$ at points $M, N$ intersecting at point $Q$. Find the range of $|O Q|$ as point $P$ moves on the unit circle.