An Area Ratio Range with a Centroid

A line through the fixed point $(1,0)$ meets the parabola $y^2=2x$ at two points $A$ and $B$. Let $C$ be the point for which the origin $O$ is the centroid of $\triangle ABC$ (that is, $C=-(A+B)$). The lines $CA$ and $CB$ meet the $y$-axis at points $M$ and $N$ respectively. Let $S_1$, $S_2$ be the areas of $\triangle CMN$, $\triangle AOB$. Find the range of $\dfrac{S_1}{S_2}$.