6. Let the two foci of the ellipse $\Gamma$ be $F_{1}, F_{2}$, and a line passing through point $F_{1}$ intersects $\Gamma$ at points $P, Q$. If $\left|P F_{2}\right|=$ $\left|F_{1} F_{2}\right|$, and $3\left|P F_{1}\right|=4\left|Q F_{1}\right|$, then the ratio of the minor axis to the major axis of the ellipse $\Gamma$ is $\qquad$.
Let $P Q$ form an angle $\theta$ with the negative direction of the $x$-axis.  
It is easy to get $\left|P F_{1}\right|=\frac{e p}{1+e \cos \theta},\left|Q F_{1}\right|=\frac{e p}{1-e \cos \theta}$, where $p=\frac{a^{2}}{c}-c=\frac{b^{2}}{c}$. If $\left|P F_{1}\right|=\lambda\left|Q F_{1}\right| \Rightarrow \lambda(1+e \cos \theta)=1-e \cos \theta \Rightarrow e \cos \theta=\frac{1-\lambda}{1+\lambda}$. Substituting $\lambda=\frac{4}{3} \Rightarrow e \cos \theta=-\frac{1}{7} \Rightarrow\left|P F_{1}\right|=\frac{\frac{b^{2}}{a}}{1-\frac{1}{7}}=\frac{7 b^{2}}{6 a}=2 a-2 c$ $\Rightarrow 7\left(a^{2}-c^{2}\right)=12 a^{2}-12 a c \Rightarrow 5 a^{2}-12 a c+7 c^{2}=0 \Rightarrow e=\frac{c}{a}=\frac{5}{7}$. Therefore, the ratio of the minor axis to the major axis of the ellipse $\Gamma$ is $\frac{b}{a}=\sqrt{1-e^{2}}=\frac{2 \sqrt{6}}{7}$.