Two Viewing-Angle Conditions

Consider the circle $O \colon x^2 + y^2 = 4$ and the line $l \colon y = kx + 5$.

1. If there exist a point $A$ on $l$ and a point $B$ on the circle $O$ with $\angle OAB = \dfrac{\pi}{6}$, find the range of possible values of $k$. 2. If for every point $A$ on $l$ there exists a point $B$ on the circle $O$ with $\angle OBA = \dfrac{\pi}{6}$, find the range of possible values of $k$.