Consider the following set of binary strings.

$S=(1({01}^{*}0{)}^{*}{10}^{*}{)}^{*}$

$($a$)$ List all the binary strings in $S$ of length at most $4.$

$($b$)$ Show that $1({01}^{*}0{)}^{*}{10}^{*}$ is unambiguous.

$($c$)$ Assuming $S$ is unambiguous, and using the length of a string as its weight, determine

the generating function for the set $S.$ Express it as a simplified rational expression.

$($d$)($Extra Credit: $1$ mark$)$ Show every string in $S$ is the binary representation of an integer

that is a multiple of $3.($A binary string $a_n{a}_{n-1}...a_0$ represents the integer ${\displaystyle \underset{i=0}{\overset{n}{}}}{a}_i{2}^i.)$