2. In the ellipse $\Gamma$, $A$ is one endpoint of the major axis, $B$ is one endpoint of the minor axis, and $F_{1}, F_{2}$ are the two foci. If $\overrightarrow{A F_{1}} \cdot \overrightarrow{A F_{2}}+\overrightarrow{B F_{1}} \cdot \overrightarrow{B F_{2}}=0$, then the value of $\frac{|A B|}{\left|F_{1} F_{2}\right|}$ is $\qquad$.
Let $A(a, 0), B(0, b), F_{1}(-c, 0), F_{2}(c, 0)$, then
$$
\overrightarrow{F_{1} A}=(a+c, 0), \overrightarrow{F_{2} A}=(a-c, 0), \overrightarrow{F_{1} B}=(c, b), \overrightarrow{F_{1} B}=(-c, b),
$$

Thus $\overrightarrow{A F_{1}} \cdot \overrightarrow{A F_{2}}+\overrightarrow{B F_{1}} \cdot \overrightarrow{B F_{2}}=(a+c)(a-c)-c^{2}+b^{2}=a^{2}+b^{2}-2 c^{2}=0$
$$
\Rightarrow a^{2}+b^{2}=2 c^{2} \Rightarrow|A B|^{2}=\frac{1}{2}\left|F_{1} F_{2}\right|^{2} \Rightarrow \frac{|A B|}{\left|F_{1} F_{2}\right|}=\frac{\sqrt{2}}{2}
$$