The sequence of traffic lights for drivers is detailed below. Each pattern has been assigned a specific label, as illustrated in Figure $1.$

The traffic lights for drivers are represented as follows: $\text{'}$R$\text{'}$ for the red light, $\text{'}$A$\text{'}$ for the amber light, and $\text{'}G\text{'}$ for the green light. When a light is ON$,$ it is denoted as $\text{'}1\text{'},$ and when it is OFF, it is denoted as $\text{'}0\text{'}.$

$($a$)$ Arrange the light outputs in the RAG order and draw the state transition diagram of the system. Assume that the transition is automatic and only triggered by the system clock.

$($b$)$ Based on the state transition diagram in part $($a$),$ draw the state table. Label state $\text{'}$a$\text{'}$ as $\text{'}{00}^{\text{'}},$ state $\text{'}$b$\text{'}$ as $\text{'}01\text{'},$ state $\text{'}$c$\text{'}$ as $\text{'}{10}^{\text{'}},$ and state $\text{'}$d$\text{'}$ as $\text{'}11\text{'}.$

$($c$)$ Construct the state assignment table. To avoid confusion, use $\text{'}x\text{'}$ and $\text{'}Y\text{'}$ to represent the

present state and $\text{'}T\text{'}$ and $\text{'}U\text{'}$ to represent the next state. Follow the given layout for the table.

$($d$)$ Draw the Karnaugh maps for the next state and the outputs, and from these, derive their respective Boolean expressions.

$($e$)$ Implement the sequential system using D flip$-$flops and appropriate logic gates. Draw a clear and well$-$labeled circuit diagram of the system.