This circuit design utilizes AND, OR$,$ and XOR gates to perform the addition of two $2-$bit binary numbers $($A and B$)$ and generate the $2-$bit sum $($C$)$ along with a carry bit $($Cout$)$ if the sum exceeds $1$ bit.

$`1.$ Include the truth table

$2.$ Include the circuit diagram, for the arithmetic circuit to illustrate its functioning.

$3.$ Provide a step$-$by$-$step explanation of how the inputs are processed through the gates to obtain the final output."

$4.$Analyze how the circuit behaves with various inputs, then classify the potential results using binary arithmetic principles.