Find the grammars for the alphabet $=\{a,b\}$ that generate the sets of:

$($a$)$ all strings with exactly one $a.$

$($c$)$ all strings with no more than three $a\text{'}$s$.$

$($b$)$ all strings with at least one $a.$

$($d$)$ all strings with at least three $a\text{'}$s$.$

In each of these cases, give convincing arguments that the grammar you give does indeed generate the indicated language.