【Example 5】How many vertices does the cube $\{(x, y, z): 0 \leqslant x \leqslant a, 0 \leqslant y \leqslant a, 0 \leqslant z \leqslant a\}$ have? Write down their coordinates respectively.

The answer to this problem is obvious, but our current goal is to verify this answer from the perspective of repeated permutations. This viewpoint and method of discussion will be very useful for discussing the properties of solids in $n$-dimensional space in the future.