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$1.[15$ points$]$ Determine what the Turing machine in Example $9\mathrm{.}7$ does when presented with the input aaabbbb.

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The set of transitions for the Turing Machine in Example $9\mathrm{.}7$ is:

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$($q$0,$a$)=($q$1,$x$,$R$),$ $($q$1,$a$)=($q$1,$a$,$R$),$ $($q$1,$y$)=($q$1,$y$,$R$),$ $($q$1,$b$)=($q$2,$y$,$L$)$

$($q$2,$ y$)=($q$2,$y$,$L$),$ $($q$2,$a$)=($q$2,$a$,$L$),$ $($q$2,$x$)=($q$0,$x$,$R$),$

$($q$0,$y$)=($q$3,$y$,$R$),$ $($q$3,$ y$)=($q$3,$y$,$R$),$ $($q$3,)=($q$4,,$R$).$