Question 125
In ellipse $\Gamma$, $A$ is an endpoint of the major axis, $B$ is an endpoint of the minor axis, and $F_1$, $F_2$ are the foci. If $\overrightarrow{AF_1} \cdot \overrightarrow{AF_2} + \overrightarrow{BF_1} \cdot \overrightarrow{BF_2} = 0$, then find the value of $\tan \angle ABF_1 \cdot \tan \angle ABF_2$.