3. Given that $\alpha-l-\beta$ is a dihedral angle of size $45^{\circ}$, $C$ is a fixed point inside the dihedral angle, and the distances from $C$ to the half-planes $\alpha, \beta$ are $2, 6$ respectively. $A, B$ are moving points in the half-planes $\alpha, \beta$ respectively. Then the minimum value of the perimeter of $\triangle A B C$ is $\qquad$
3. $4 \sqrt{10+3 \sqrt{2}}$.

Construct the symmetric points $P, Q$ of point $C$ with respect to planes $\alpha, \beta$. It is easy to prove that when $A, B$ are the intersection points of line $P Q$ with planes $\alpha$, $\beta$ respectively, the perimeter of $\triangle A B C$ is minimized, and this minimum perimeter value is
$$
|P Q|=\sqrt{4^{2}+12^{2}-2 \times 4 \times 12 \cos \left(180^{\circ}-45^{\circ}\right)}=4 \sqrt{10+3 \sqrt{2}} \text {. }
$$