7. Given the hyperbola $C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0)$ with an eccentricity of $\frac{\sqrt{5}}{2}, F_{1}, F_{2}$ are the left and right foci of $C$, respectively. A line $l$ passing through $F_{2}$ intersects $C$ at points $A$ and $B$ (point $A$ is in the first quadrant), and $\overrightarrow{A F_{2}}=3 \overrightarrow{F_{2} B}$. If the area of $\triangle F_{1} A B$ is $\frac{32}{3}$, then the inradius of $\triangle F_{1} A B$ is $\qquad$.