$(30$ points$)$ Objective: Design a four$-$state finite state machine $($FSM$)$ using D flip$-$flops to control a lighting system. The system adjusts the light through four distinct brightness levels: Off, Low, Medium, and High,

Description: You are required to design an FSM that controls the brightness of a light with four distinct levels. The light should adjust its brightness based on a single input, cycling through the states in a predefined manner.

States and Encodings:

Off $(0)$:$00$

Low $($L$)$: $01$

Medium $($M$)$: $10$

High $(H)$:$11$

Input:

A single binary input $($Toggle Input$)$:

$\text{'}1\text{'}-$ If the input is $\text{'}1\text{'},$ the FSM should increase the brightness level sequentially from Off to Low, from Low to Medium, from Medium to High.

$\text{'}0\text{'}-$ If the input is $\text{'}0\text{'},$ the FSM should transition to the Off state regardless of the current state.

Special Note: From the High state, no matter what the input is$,$ the light should transition directly back to Off. This includes both $\text{'}0\text{'}$ and $\text{'}1\text{'}$ as inputs.

Requirements:

State Transition Table: Create a detailed table that maps the current state and input to the next state and output $($brightness level$).$

State Transition Diagram: Illustrate the transitions between states visually in a diagram, showing how the input affects the state changes, especially highlighting the transition from High directly back to Off.

Logic Equations: Derive logic equations for the $D$ flip$-$flops' inputs $($D$1,$ D$0)$ based on the transitions outlined in the table.

Circuit Design: Design a schematic diagram using two $D$ flip$-$flops and the necessary logic gates to implement the described FSM$.$