A Chord Through a Shared Vertex Pair

Let $A$ and $B$ be the common vertices of the ellipse $\dfrac{x^2}4+\dfrac{y^2}3=1$ and the hyperbola $\dfrac{x^2}4-\dfrac{y^2}3=1$. Let $M$ be a point on the hyperbola; the lines $MA$ and $MB$ meet the ellipse again at points $C$ and $D$ respectively. If the line $CD$ passes through a focus $F$ of the ellipse, find the length of the segment $CD$.