Equal Segments on a Focal Vertical

Through the left focus $F_1$ of the ellipse $W:\dfrac{x^2}{2}+y^2=1$, a line $l_1$ meets the ellipse at points $A$ and $B$, where $A(0,1)$. A second line $l_2$ through $F_1$ meets the ellipse at points $C$ and $D$ (distinct from $A$ and $B$), with $D$ not equal to $(0,-1)$. The vertical line through $F_1$ meets the lines $AD$ and $BC$ at points $E$ and $G$ respectively.

1. Find the coordinates of $B$ and the equation of the line $l_1$. 2. Prove that $|EF_1|=|GF_1|$.