6. As shown in the figure, in the right triangular prism $A B C-A_{1} B_{1} C_{1}$, $\angle A C B=90^{\circ}, A C=6, B C=C C_{1}=\sqrt{2}, P$ is a moving point on the line segment $B_{1} C$. Then the minimum value of $A P+P C_{1}$ is $\qquad$ .
6. $5 \sqrt{2}$ Detailed Explanation: As shown in the figure, connect $A B_{1}$, and unfold $\triangle C_{1} B_{1} C$ along $B_{1} C$ onto the plane $A B_{1} C$. Then, the minimum value of $A P+P C_{1}$ is the length of segment $A C_{1}$. In $\triangle A C C_{1}$, $A C=6, C C_{1}=\sqrt{2}, \angle A C C_{1}=\angle A C B_{1} + \angle B_{1} C C_{1}=135^{\circ}$. By the cosine rule, we get $A C_{1}=\sqrt{6^{2}+(\sqrt{2})^{2}-2 \times 6 \sqrt{2} \cos 135^{\circ}}=5 \sqrt{2}$.