7. In the Cartesian coordinate system $x O y$, $F_{1}$ and $F_{2}$ 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 $A$ and $B$, respectively. If $F_{2} B=A B$, then the value of $b$ is $\qquad$.
$7.1+\sqrt{3}$.
From the known information and the definition of a hyperbola, we get
$$
\begin{array}{l}
A F_{1}=B F_{1}-A B=B F_{1}-B F_{2}=2, \\
A F_{2}=2+A F_{1}=4 .
\end{array}
$$

As shown in Figure 3, let $A B$ be tangent to the circle at point $T$.