(16) (This sub-question is worth 20 points)

Given the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ with eccentricity $e=\frac{1}{2}$, the endpoints of the minor axis are $B_{1}, B_{2}$, and the foci are $F_{1}, F_{2}$. The radius of the incircle of the quadrilateral $F_{1} B_{1} F_{2} B_{2}$ is $\frac{\sqrt{3}}{2}$.
(I) Find the equation of the ellipse $C$;
(II) A line passing through the left focus $F_{1}$ intersects the ellipse at points $M, N$ and the line $x=-4$ at point $P$. Let $\overrightarrow{P M}=\lambda \overrightarrow{M F_{1}}, \overrightarrow{P N}=\mu \overrightarrow{N F_{1}}$. Prove that $\lambda+\mu$ is a constant, and find this constant.