A Ratio of Two Symmetric Chords

Let $A$ and $B$ be two points on the ellipse $\dfrac{x^2}4+\dfrac{y^2}2=1$, symmetric about the origin $O$, and let $P$ be a point on the ellipse $\dfrac{x^2}{12}+\dfrac{y^2}6=1$. The segments $PA$ and $PB$ meet the ellipse $\dfrac{x^2}4+\dfrac{y^2}2=1$ again at points $M$ and $N$ respectively. If $\overrightarrow{AB}=t\overrightarrow{MN}$, find $t$.