6. As shown in the figure, in the right triangular prism $A B C-A_{1} B_{1} C_{1}$, $\angle A C B=90^{\circ}, B C=C C_{1}=2, A C=4 \sqrt{2}, P$ is a point on $B C_{1}$, then the minimum value of $C P+P A_{1}$ is
As shown in the figure, the right triangular prism $A B C-A_{1} B_{1} C_{1}$ is extended to form a rectangular prism $A C B D-A_{1} C_{1} B_{1} D_{1}$. In the plane $A_{1} C_{1} B$, an isosceles right triangle $\triangle B C^{\prime} C_{1}$ is constructed outside the rectangle $A_{1} D B C_{1}$, with $B C_{1}$ as one of its sides. Connect $C^{\prime} A_{1}$.
Since $\triangle B C^{\prime} C_{1} \cong \triangle B C^{\prime} C_{1} \Rightarrow C P=C^{\prime} P$,
then $C P+P A_{1}=C^{\prime} P+P A_{1} \geqslant C^{\prime} A_{1}$,
thus the length of $C^{\prime} A_{1}$ is the desired value.
In $\triangle A_{1} C_{1} C^{\prime}$, $C^{\prime} A_{1}^{2}=(4 \sqrt{2})^{2}+2^{2}+2 \cdot 4 \sqrt{2} \cdot 2 \cdot \frac{\sqrt{2}}{2}=52$, so the minimum value of $C P+P A_{1}$ is $2 \sqrt{13}$.