In this exercise we will explore designing an integrated circuit for use within a coffee grinder. Rather than tackling the entire system, we'll focus on a specific aspect of it$.$ Imagine the coffee maker already has a circuit responsible for managing the heating element and the water pump. Your objective is to design a simple circuit to control the coffee grinder. The grinder is driven by a motor with two input bits:

G: If G$1,$ the grinder operates. If G$0,$ the grinder is inactive.

F: If F$1$ and G$1,$ the grinder operates in a fine setting. If F$0$ and G$1,$ the grinder operates in a coarse setting.

The manufacturer of this coffee maker requests a synchronous circuit that will control the grinder to switch between fine and coarse grinding each time the coffee maker is turned on$.$ The circuit to be designed will receive a single input bit I from the circuit controlling the heating element.

When I$1,$ the grinder should operate.

When I$0,$ the grinder should remain inactive.

To guide your design process, complete parts $($a$)$ to $($d$)$ below.

$($a$)$ Draw a Finite State Machine that describes this state circuit. Since this circuit has $4$ possible combinations of outputs, it should have $4$ states. Be sure to include the

inputs and outputs in the FSM$.$ Order the outputs as GF$.$ Therefore, a transition in

the FSM will be labeled as I$$GF$.$ The change in grinding setting should be triggered

by the coffee maker turning on$.$

$($b$)$ Create a truth table to describe the inputs, outputs, and transitions of your FSM$.$ That is$,$ create a truth table describing the combinational logic of your eventual state circuit. Since there are $4$ states, you must have $2$ bits of $$input$$ to encode the current state, and $2$ bits of $$output$$ to encode the next state. $($Hint: your table should have $8$ rows and $7$ columns.$)$