Consider a finite state transducer (FST) with $Q=\left\{q_{0}, q_{1}\right\}, \Sigma=\{0,1\}, \Gamma=\{a, b$, $\mathrm{c}\}$, initial state $\mathrm{q}_{0}$, and the following transitions:
$$
\begin{array}{ll}
\delta\left(q_{0}, 0\right)=q_{1}, & \delta\left(q_{0}, 1\right)=q_{0} \\
\delta\left(q_{1}, 0\right)=q_{0}, & \delta\left(q_{1}, 1\right)=q_{1}
\end{array}$$
and output function:
$$\begin{array}{ll}
\theta\left(q_{0}, 0\right)=a, & \theta\left(q_{0}, 1\right)=c \\
\theta\left(q_{1}, 0\right)=b, & \theta\left(q_{1}, 1\right)=a
\end{array}$$
What will be the output string for the input $1010$?
- $\text{abac}$
- $\text{caba}$
- $\text{caab}$
- $\text{acab}$
