$(12$ points$)$ Let $=\{a\}$ and $L_1=\{a^{3n+1}$:$n0\}.($These are stringe of $3n+1$ a $)$

$($a$)(4$ points$)$ Find two strings that are in $L_1$ and two strings that are not:

$($b$)(4$ points$)$ Give the state diagram of a DFA $M$ which recognizes $L_1.$

$($c$)(4$ points$)$ Give a $($short$)$ justification for wly $M$ accepts a string $w$ if and only if $win{L}_1.($There are two claims to justify here if $M$ accepts $w,$ then $win{L}_1,$ and if $win{L}_1$ then $M$ accepts $w.)$

No AI please