Let $L$ be the language over the alphabet $\{a,b\}$ that consists of strings that have equal many $a\text{'}$s and $b\text{'}$s$.$ Consider a grammar for $L$ with one rule:

$S|ba|bSa|aSb|SS$

$($a$)$ The string aababbba can be derived from $S$ in $5$ steps. Show these steps.

$S$

$($b$)$ Which of the following languages are regular $($list all that are$)$:

$L{a}^{**}{b}^{**},L{a}^{**}{b}^{**},L*a^{**}{b}^{**},L^{**}$

$($c$)$ Is $L$ countable? $($Answer with: Yes or No$.)$

$($d$)$ Modify the grammar for $L$ to write a grammar for the language over $\{a,b,c,d\}$ that consists of all strings which the total number of $a^{\text{'}}s$ and $b^{\text{'}}s$ is equal to the total number of $c^{\text{'}}s$ and $d^{\text{'}}s.$