7. In the Cartesian coordinate system $x O y$, $F_{1}, F_{2}$ are the left and right foci of the hyperbola $x^{2}-\frac{y^{2}}{b^{2}}=1(b>0)$, respectively. A tangent line is drawn from point $F_{1}$ to the circle $x^{2}+y^{2}=1$, intersecting the left and right branches of the hyperbola at points $A, B$, respectively. If $F_{2} B=A B$, then the value of $b$ is $\qquad$.
Solve:
As shown in the figure, $c=\sqrt{b^{2}+1}, F_{1} B-F_{2} B=F_{1} A=2$,
$$
F_{2} A-F_{1} A=2 \Rightarrow F_{2} A=4 \text {, }
$$

In $\triangle O F_{1} T$, $\sin \alpha=\frac{1}{c} \Rightarrow \cos \alpha=\frac{b}{c}$.
Thus, in $\triangle A F_{1} F_{2}$,
$$
\begin{array}{l}
16=4+4 c^{2}-2 \cdot 2 \cdot 2 c \cdot \frac{b}{c} \\
\Rightarrow b^{2}-2 b-2=0 \Rightarrow b=\sqrt{3}+1 .
\end{array}
$$