A Dihedral Over a Frustum

In a regular triangular frustum $ABC\text{-}A_1B_1C_1$, $AB = 2A_1B_1 = 2\sqrt{3}$ and $AA_1 = 2$. Let $D$ be the midpoint of edge $AB$, and let $l$ be a moving line in the plane $A_1B_1C_1$. If $\theta$ denotes the dihedral angle $C\text{-}l\text{-}D$, find the minimum value of $\cos\theta$.