6、$F_{1}, F_{2}$ are the two foci of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$, and $P$ is a point on the ellipse. If the area of $\Delta P F_{1} F_{2}$ is $1$, $\tan \angle P F_{1} F_{2}=\frac{1}{2}$, and $\tan \angle P F_{2} F_{1}=-2$. Then $a=\frac{\sqrt{15}}{2}$.