$3 \cdot 3$ Label the endpoints of a diameter of a circle with the number 1, then equally divide each of the resulting semicircles, and write the sum of the numbers at the endpoints of the two semicircles at their division points (first step). Then equally divide the 4 arcs obtained, and write the sum of the numbers at the endpoints of these arcs at their division points (second step). Find the sum of all numbers marked on the circle after performing this process $n$ times.
[Solution] Let the sum of all numbers after the $k$-th step be denoted as $S_{k}$, then $S_{0}=2, S_{1}=$ 6. During the $(k+1)$-th step, since each number in $S_{k}$ is at the endpoint of exactly two arcs, each number is used twice, so we have $S_{k+1}=S_{k}+2 S_{k}=3 S_{k}$. By this recurrence relation, we get
$$
S_{n}=3 S_{n-1}=3^{n} S_{0}=2 \times 3^{n},
$$

which means after $n$ steps, the sum of all numbers marked on the circle is $2 \times 3^{n}$.