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.