$6.$B$\mathrm{.}2.[5$ marks$]$ A set $L$ of bit strings is defined recursively as follows.

Basis step: $inL.$

Recursive step: If winL, then $w01$inL, and $1w0$inL.

$($i$)$ List all the strings in $L$ that have length $4.$

$($ii$)$ For a string $w,$ let $P(w)$ be the statement "The number of $0\text{'}$s in $w$ is equal to the number of $1\text{'}$s in $w."$ Use structural induction to prove that $P(w)$ is true for every string $w$ in $L.$ Include all the necessary details.