As shown in the figure, let the left focus of the hyperbola be $F_{1}$. Since $|A P|+|P F|=|A P|+2 a+\left|P F_{1}\right|$ $\geqslant 2 a+\left|A F_{1}\right|$, the equality holds when $A, P, F_{1}$ are collinear. Therefore, the minimum perimeter of $\triangle A P F$ is $2 a+\left|A F_{1}\right|+|A F|=3 \sqrt{2}+\sqrt{10}$.
6. The hyperbola $C: x^{2}-y^{2}=2$ has its right focus at $F$, and $P$ is any point on its left branch. Point $A$ has coordinates $(-1,1)$. Then the minimum perimeter of $\triangle A P F$ is $\qquad$.