Three. (20 points) In a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length $a$, $X$ is the center of the square $A A_{1} B B_{1}$, $Y$ is the center of the square $B B_{1} C_{1} C$, and $Z$ is on the diagonal $B D$ such that $D Z=3 Z B$. Find the area of the section passing through $X, Y, Z$.

Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
Three, first find the section. Draw the projections $X', Z'$ of $X, Z$ on the plane $B B_{1} C_{1} C$. The two parallel lines $Z Z'$ and $X X'$ determine a plane. Connect $X Z$ and $X' Z'$, intersecting at $O$, then $O$ is on the plane $B B_{1} C_{1} C$. Connect $O Y$ intersecting $B C$ at $P$ and $B_{1} C_{1}$ at $S$. Since $X'$ is the midpoint of $B B_{1}$ and $Z' C = 3 Z' B$, it follows that $P$ is the midpoint of $B C$, and thus $S$ is the midpoint of $B_{1} C_{1}$. Connect $P Z$ intersecting $A B$ at $Q$, then $Q$ is the midpoint of $A B$.

Connect $Q X$ intersecting $A_{1} B_{1}$ at $R$, then $R$ is the midpoint of $A_{1} B_{1}$.

Connect $R S$, then the rectangle $P Q R S$ passes through $X, Y, Z$, which is the desired section.

At this time, $P Q = \frac{\sqrt{2}}{2} a$, $Q R = a$, and the area of the section is $\frac{\sqrt{2}}{2} a^{2}$.