$(30$ points$)$ Given the reaction system $A=(S,A)$ with background set $S=\{1,2,3,4\}$ and a set of reactions

$A=\{a,b,c,d\},$ where $a=(\{1\},\{2\},\{2\}),b=(\{2\},\{1\},\{1\}),$

$c=(\{3\},\{4\},\{3\}),$ and $d=(\{4\},\{3\},\{4\}),$ denote the power set of $S$ by $P(S).$

$($a$)$ For every set TsubeS check whether each reaction is enabled in $T$ and list all reactions that are enabled in $T.$

$($b$)$ For every set TsubeS find $re{s}_A(T).$ Show your work.

$($c$)$ Draw the zero$-$context graph $G_A^0=(V,E)$ for the reaction system $A=(S,A)$ with vertices $V=P(S)$ and

a set of directed edges $E=\{(T,re{s}_A(T))|TsubeS\}.($Your graph diagram should have nonintersecting edges

and the components, if more than one, should be clearly separated.$)$