Problem $1$: You are tasked with designing a Moore machine for an automatic toilet.

The toilet should implement the following operations:

i$)$ The system has one input: a sensor detecting whether a person is present or not.

ii$)$ The system has one output: a signal that will trigger the toilet's flushing mechanism.

iii$)$ Additionally, the system should have a reset signal to return it to its default "idle" state.

When empty, the toilet should remain in an idle state. When a person enters the toilet, the system should transition to an in use state. The toilet should remain in this state as long as a person is present. Once the person exits the toilet, the system should transition to the flush state. The system should be able to handle the cases when a single person is present or when there is a cue $($that is$,$ one person uses the facility after another$).$

$1)$ How many D flip flops do we need to implement this logic? $(1$ point$).$

$3)$ Draw the state machine diagram that would implement this automatic toilet $(3$ points$).$

$4)$ Using Karnaugh maps, derive the boolean algebra equation for the output logic of this system $(1$ points$).$

$5)$ Using Karnaugh maps, derive the boolean algebra equation for the input logic of this system $(2$ points$).$

$6)$ Draw the full circuit that would implement this device. For the input logic, use a decoder $(3$ points$).$