10. (10 points) As shown in the figure, there is a beautiful fountain in the center of the square. Xiao Ming starts from point $A$, walks around the fountain along the small path without repeating, and finally returns to point $A$. The number of ways to walk around the fountain once is $\qquad$. (The two circles of the fountain and the line segments between the two circles in the figure all represent paths, and walking around the fountain once means that the walking route is a closed route and the fountain is inside this route)
【Solution】Solution: If only considering the two clockwise line segments $a_{1}, a_{2}$, only one can be chosen. Similarly, for each pair of line segments $b_{1}, b_{2}, c_{1}, c_{2}, \cdots, h_{1}, h_{2}$, only one can be chosen. Once 8 line segments are selected, the path of the loop is uniquely determined, giving a total of $2^{8}=256$ ways. If both $a_{2}$ and $h_{2}$ are chosen, the line segment $K$ at point $A$ would be traversed twice, which is not allowed.
$\therefore$ The number of ways to choose both $a_{2}$ and $h_{2}$ is $2^{6}=64$,
$\therefore$ The number of clockwise ways is $2^{8}-2^{6}=192$,
By symmetry, the number of ways for counterclockwise is the same as for clockwise,
$\therefore$ The total number of ways is $192 \times 2=384$.
Therefore, the answer is: 384.