A Geometric Focal Progression

The hyperbola $C:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ ($a>0$, $b>0$) has left and right foci $F_1$, $F_2$, eccentricity $3$, and the two intersection points of the line $y=2$ with $C$ are at distance $\sqrt 6$ from each other.

1. Find $a$ and $b$. 2. A line $l$ through $F_2$ meets the left and right branches of $C$ at points $A$ and $B$ respectively, with $|AF_1|=|BF_1|$. Prove that $|AF_2|$, $|AB|$, $|BF_2|$ form a geometric progression.