Give a regular description for the image of the language $L=\{ w \in \{a,b\}^* \mid |w|_a\in 2\mathbb{N}\}$ through this transducer: 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 $baa\in L$ and the following step-by-step execution:

• $0~~~baa$
  Initially, the transducer is at state $0$ and it remains to read the whole word $baa$. Moreover, no output has been produced yet.
• $bb~~~2~~~aa$
  By reading $b$ at state $0$, the transducer has moved to state $2$ and outputted $bb$.
  (Equivalently, the 2nd rewrite rule has been applied.)
• $bb~a~~~1~~~a$
  By reading $a$ at state $2$, the transducer has moved to state $1$ and outputted $a$.
  (Equivalently, the 5th rewrite rule has been applied.)
• $bb~a~b~~~2$
  By reading $a$ at state $1$, the transducer has moved to state $2$ and outputted $b$.
  (Equivalently, the 3rd rewrite rule has been applied.)

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