Draw state diagrams for deterministic finite$-$state transducers over $\{$a$,$ b$\}$

that do the following.

$($i$)$ On input w$,$ produce output an$,$ where n is the number of occurrences

of the substring ab in w$.$

$($ii$)$ On input w$,$ produce output an$,$ where n is the number of occurrences

of the substring aba in w$.$

$($iii$)$ On input w$,$ produce a string of length w whose ith symbol is an a if

i $=1,$ or if i $>1$ and the ith and $($i $-$ l$)$st symbols of ware different;

otherwise, the ith symbol of the output is a b$.$ For example, on input

aabba the transducer should output ababa, and on input aaaab it should

output abbba.