Give a regular description for the image of the language $L=\{ w \in \{a,b\}^* \mid |w|_a\in\dot{2}\}$ through the transducer represented here. Such transducer can be alternatively formalized as the following set of rewrite rules: In this alternative setting, to rewrite a word $w\in L$ we would start the execution with the word $0w$ and proceed until obtaining a word of the form $w'0$ or $w'1$ or $w'2$ (i.e., a word that cannot be rewritten any further), where $w'$ would be the generated output. In order to get more intuition on how this transducer works, consider the word $abba\in L$ and the following execution:

• $0~~abba$
  Initially, the transducer is at state $0$ and it remains to read the whole word $abba$. Moreover, no output has been produced yet.
• $aba~~1~~bba$
  By reading an $a$ from state $0$, the transducer has moved to state $1$ and outputted $aba$. (Equivalently, the 1st rewrite rule has been applied.)
• $aba~a~~0~~ba$
  By reading a $b$ from state $1$, the transducer has moved to state $0$ and outputted an $a$. (Equivalently, the 4th rewrite rule has been applied.)
• $aba~a~bb~~2~~a$
  By reading a $b$ from state $0$, the transducer has moved to state $2$ and outputted $bb$. (Equivalently, the 2nd rewrite rule has been applied.)
• $aba~a~bb~a~~1$
  By reading an $a$ from state $2$, the transducer has moved to state $1$ and outputted an $a$. (Equivalently, the 5th rewrite rule has been applied.)

Since there is no more input, the execution terminates. Hence, the image of the word $abba\in L$ through the transducer is $abaabba$.