5. Given a right triangular prism $A B C-A_{1} B_{1} C_{1}$ with a right-angled triangle base, $\angle A C B=90^{\circ}, A C=$ $6, B C=C C_{1}=\sqrt{2}, P$ is a moving point on $B C_{1}$, then the minimum value of $C P+P A_{1}$ is
As shown in the figure, construct the rectangular prism $A C B D-A_{1} C_{1} B_{1} D_{1}$,  
fold $\triangle B C C_{1}$ onto the diagonal face $A_{1} C_{1} B D$, becoming $\triangle B C^{\prime} C_{1}$. Connect $C^{\prime} A_{1}$, intersecting $B C_{1}$ at point $P$, at this moment $C P+P A_{1}=C^{\prime} A_{1}$ is the minimum. Since $\angle C^{\prime} C_{1} A_{1}=135^{\circ}$, applying the cosine rule in $\triangle C^{\prime} C_{1} A_{1}$ yields $C^{\prime} A_{1}^{2}=2+36+2 \cdot \sqrt{2} \cdot 6 \cdot \frac{\sqrt{2}}{2}=50 \Rightarrow C^{\prime} A_{1}=5 \sqrt{2}$. Therefore, the minimum value of $C P+P A_{1}$ is $5 \sqrt{2}$.