Problem $3(10$ points$)$ Consider the Turing machine:

$M=(Q,A,T,,q,$ accept, neyect $)$

such that:

$Q=\{q,r,s,t,u,$accept,reject$\}$;

$A=\{a,b\}$;

$T=\{B,a,b\}$;

and $$ is defined by the following transition set:

$[q,a,r,b,R]$

$[q,b,s,b,R]$

$[q,B,q,b,R]$

$[r,a,r,a,R]$

$[r,b,s,a,R]$

$[r,B,t,B,L]$

$[s,a,s,a,R]$

$[s,b,r,a,R]$

$[s,B,u,B,L]$

$[t,a,t,a,L]$

$[t,b,$accept,$a,L]$

$[t,B,$reject,$B,L]$

$[u,a,t,a,L]$

$[u,b,$reject,$a,L]$

$[u,B,$accept,$B,L]$

$($a$)$ Write a regular expression that defines the set of strings on which $M$ diverges. If such a regular expression does not exist, prove it$.$

$($b$)$ Write a regular expression that defines the set $L$ of strings that $M$ accepts. If such a regular expression does not exist, prove it$.$

$($c$)$ Write a regular expression that defines the set of strings on which $M$ halts and rejects. If such a regular expression loes not exist, prove it$.$

Does $M$ decide the language $L$ of part $($b$)?$ Answer Yes or No$.$