# Specify a decisive Turing machine that calculates the following function $f$.

Specify a decisive Turing machine that calculates the following function $$f$$: $$\small f:\{a,b\}^*\to\{a,b\}^*\textrm{ with } f(w)= \begin{cases} (bba)^{3\cdot\#_b(w)}& \text{if } \#_a(w) \text{ not devidable by }4\\ \text{undefined} & \text{otherwise.} \end{cases}$$ where $$\#_b(w)$$ is the amount of $$b$$'s in $$w$$.

Let $$M=(K,\Sigma,\Gamma,\delta,s,F) \text{ with } s\in K$$ initial state where $$K$$ is the set of states; $$\Sigma$$ is the alphabet; $$\Gamma$$ is the infinite tape; $$F\subseteq K$$ are final states and $$\delta:K\times\Gamma\to K\times \Gamma \times \{R,N,L\}$$ is the transition function with ($$R$$=right, $$L$$=left, $$N$$=neutral).

I have no clue how to go on and design the TM in order to calculate $$f$$. Can you give me some hints?

Hint: We need 4 states $$s_0,\dots, s_3$$ to track $$\#_a(w)$$ mod $$4$$, such that only $$s_0$$ is not final, and we have to output $$bbabbabba$$ whenever we meet letter $$b$$.