Again Multi-Peg Tower of Hanoi

In CSE courses, we frequently cite the Tower of Hanoi problem to make the concept of recursive procedures clear. In an ordinary Tower of Hanoi problem, there are three pegs $S_1$, $S_2$, and $S_3$. Initially, circular disks with holes are mounted on $S_1$ with smaller disks on top of larger ones. All the disks are of different diameters. The task is to shift all the disks to $S_3$, always shifting