$$For a given language below, construct a TM with a single final state that accepts it$.$

$(1)[10]$ L $=\{$w $|($ab$)$n $|$ n $^31\}$ where S $=\{$a$,$ b$\}.$

$(2)[10]$ L $=\{$w $||$w$|$ is a multiple of $4\}$ where S $=\{$a$,$ b$\}.$

Note: $|$$|=0$ is also a multiple of $4.$ So$,$ in L$.$

$(3)[10]$ L$($a$($a$+$b$)*)$where S $=\{$a$,$ b$\}.$ Hint: the number of states is not more than three.

$(4)[10,$ optional$]$ L $=\{$w $|$ anb$2$n $|$ n $^31\}$ where S $=\{$a$,$ b$\}.$