13. In $\triangle A B C$, $\angle B=\frac{\pi}{4}, \angle C=\frac{5 \pi}{12}, A C=2 \sqrt{6}, A C$'s midpoint is $D$. If a line segment $P Q$ of length 3 $(P$ is to the left of $Q$) slides on line $B C$, then the minimum value of $A P+D Q$ is
Given, $\angle A=\frac{\pi}{3} \Rightarrow BC=\frac{AC \sin A}{\sin B}=6$, let the midpoint of $AB$ be $E$, and connect $DE, EP$. Then quadrilateral $EDQP$ is a parallelogram, so
$$
DQ=EP \Rightarrow AP+DQ=AP+EP \text{. }
$$

Construct the symmetric point of $E$ with respect to line $BC$ as $E'$, and connect $AE'$ intersecting line $BC$ at point $P'$. By $AP+EP \geqslant AP'+E'P'=AE'$, we know that the minimum value of $AP+DQ$ is $AE'$. Clearly, $AB=\frac{AC \sin C}{\sin B}=3\sqrt{2}+\sqrt{6}$, and $\angle EBE'=90^{\circ}$, so $AE'=\frac{\sqrt{5}}{2} AB=\frac{3\sqrt{10}+\sqrt{30}}{2}$.