Consider the nfa $N$ shown below. Let $L=L(N).$

$($a$)$ Find a dfa $D_1$ such that $L=L(D_1).$

$($b$)$ Find a dfa $D_2$ such that $\frac{?}{\text{b}\text{a}\text{r}}(L)=L(D_2).$

$($c$)$ Find an nfa $N^{\text{'}}$ such that $L(N^{\text{'}})=L(N)\{a^5\}.$