Question

There is a set of $1000$ switches, which are ordered in a row so that each switch is given a distinct rank from $1$ to $1000$. For example, the $i$-th switch refers to the switch given rank $i$. Each switch has four positions, called $A, B, C$, and $D$. When the position of any switch changes, it is only from $A$ to $B$, from $B$ to $C$, from $C$ to $D$, or from $D$ to $A$. Initially each switch is in position $A$. The switches are labeled arbitrarily with the $1000$ different integers $(2^x)(3^y)(5^z)$, where $x, y$, and $z$ take on the values $0, 1, \ldots, 9$. At step $i$ of a $1000$-step p