1. 26 At the ends of a diameter of a circle, the numbers 1 are marked. Then, each resulting semicircle is bisected, and the sum of the numbers at the endpoints is written at the midpoint of each arc (first step). Next, the 4 arcs obtained are bisected, and the sum of the numbers at the endpoints is written at the midpoint of each arc (second step). This process is repeated several times. Find the sum of all the numbers written.

How many steps were performed in total?
[Solution] After the first step, the sum of all numbers equals $6=2 \times 3$.
Let $S_{n}$ be the sum of all numbers after $n$ steps. It is not hard to prove that after $n+1$ steps, the sum of all numbers will be $2 S_{n}+S_{n}=3 S_{n}$.

Thus, the sum of the numbers triples with each step. Therefore, at the $n$-th step, the sum equals $2 \times$ $3^{n}$.