Let $\backslash$Sigma $=\{$a$,$ b$,$ c$\}.$ Give a finite automaton $($DFA or NFA$)$ that accepts exactly the following language:

$\{$w in $\{$a$,$ b$,$ c$\}$

$$

$|$ #a$($w$)$ is a multiple of $3$ and #b$($w$)$ arbitrary and w contains at most one c $\}$

Also briefly explain how your automaton works $($e$.$g$.$ for each state, what properties does the part of

the word read so far have?$).$

$($Note: #x$($w$)$ is the number of occurences of the symbol x in the word w$.$ A multiple also includes $0$

times.$)$