$(30$ points$)$ For the alphabet $V=\{a,b\},$ splicing rules $R=\{r_1,r_2\}$ where $r_1=a$#$ab$$$ab$#$b$ and

$r_2=a$#$$a$#$b,$ and given the language $L=\{aaabb,$aabbb$\}$:

$($a$)$ State the $H$ scheme.

$($b$)$ Show how each word in ${}_1(L)$ will form: show the splicing operation as shown in class and give ${}_1(L).$

$($c$)$ Using the construction in the proof of Theorem $1$ in the Splicing and Regularity paper, construct all

automata $M_0,M_1,$cdots,$M_k$ for the language ${}_1^{**}(L),$ starting with the language $L$ and splicing rules $R.$ Draw each

stage from $M_0$ to $M_k$ separately. What is $L(M_i)$ for each $1ik?$ What is ${}_1^{**}(L)?$