4. A plane intersects a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length 1, passing through vertex $C_{1}$, intersecting the midpoint of $A_{1} D_{1}$ at $E$, the point on $A_{1} A$ closer to $A$ at $F$, $A B$ at $G$, and $B C$ at $H$. Find the perimeter of the section $C_{1} E F G H$.
4. From question 3, we know that the section intersects the extension of edge $B B_{1}$ at $I$, then $B I=\frac{1}{2} B B_{1}$, it can be proven that $\frac{A G}{G B}=\frac{A F}{B I}=\frac{1}{2}, \frac{B H}{B_{1} C_{1}}=\frac{B I}{B_{1} I}=$ $\frac{1}{3}$, thus $B G=\frac{2}{3}, B H=\frac{1}{4}$, from which we can find $G H=\frac{\sqrt{5}}{3}, C_{1} H=\frac{\sqrt{13}}{3}, F G=\frac{5}{12}, E F=\frac{\sqrt{13}}{4}, C_{1} E=\frac{\sqrt{5}}{2}$, hence the perimeter is $\frac{5}{12}+\frac{5 \sqrt{5}}{6}+\frac{7 \sqrt{13}}{12}$.