Give a regular description for the image of the language $L=\{ xay \in \{a,b\}^* \mid |y|=1 \}$ 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 $w'1$ (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 $bbaa\in L$ and the following execution:

• $0~~bbaa$
  Initially, the transducer is at state $0$ and it remains to read the whole word $bbaa$. Moreover, no output has been produced yet.
• $b~~1~~baa$
  By reading a $b$ from state $0$, the transducer has moved to state $1$ and outputted $b$. (Equivalently, the 2nd rewrite rule has been applied.)
• $b~aa~~1~~aa$
  By reading a $b$ from state $1$, the transducer has moved to state $1$ and outputted $aa$. (Equivalently, the 4th rewrite rule has been applied.)
• $b~aa~b~~0~~a$
  By reading an $a$ from state $1$, the transducer has moved to state $0$ and outputted $b$. (Equivalently, the 3rd rewrite rule has been applied.)
• $b~aa~b~aba~~0$
  By reading an $a$ from state $0$, the transducer has moved to state $0$ and outputted $aba$. (Equivalently, the 1st rewrite rule has been applied.)

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