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 process, the $i$-th switch is advanced one step, and so are all the other switches whose labels divide the label on the $i$-th switch. after step $1000$ has been completed, how many switches will be in position $a$?