$(16$ points$)$ Consider the following device. It has two inputs, you may select $a,$ to drop

a marble, and you may select $b$ to switch the orientation of the paddle. Every time a

marble is dropped onto the paddle, it will fall into one of the two tubes depending upon

which way the paddle was facing. After a marble hits the paddle, it also automatically

flips, after the ball hits. You may also flip it manually by selecting $b.$ Let $=\{a,b\},$

and let $L$ be the language of words such that every word in $L$ corresponds to a sequence

of inputs such that the last marble dropped is accepting. Give a DFA for language

$L.$ For example, some strings in $L$ include $a,$abbbbbb,aba,abababb. Some strings not

in $L$ include $aa,bbb,$abba.