$4.$ A jukebox is controlled by a circuit to determine if it should play a selected song. The circuit outputs a $1$ under the following conditions, all of which must be true.

$$ The jukebox is in active operating mode.

$$ A legitimate song must have been selected. Songs are numbered $10-31$ but the input is a number from $10-99,$ so numbers $32-99$ are not legal.

$$ The user must have inserted payment.

$$ The record containing the song and the turntable must be free of obstruction.

The circuit receives input as follow, outputting a $1$ if the above conditions are true.

$$ Operating mode is denoted by variables m$1$m$0$ and is $00($off mode$),01($maintenance mode$),10($active mode$).11$ is not used.

$$ The song selection is converted from a $2-$digit input into $7-$bit binary number denoted by variables i$6$i$5$i$4$i$3$i$2$i$1$i$0$ where $0-31$ is a legal number.

$$ Payment is one of credit card, coins or tokens. These are indicated through three variables: CC$,$ CS$,$ TS$.$ If any of these are $1,$ the payment condition is fulfilled.

$$ Four sensors detect obstructions: R $($record rack$),$ A $($moving arm$),$ T $($turntable$)$ and N $($turntable needle$).$ The condition requires that there are no obstructions.

a$.$ Derive the Boolean expression for the circuit. Remember that the Boolean expression consists of Boolean variables $($m$1,$ m$0,$ i$6,...,$ i$0,$ CC$,$ CS$,$ TS$,$ R$,$ A$,$ T$,$ N$)$ and Boolean operators $($AND$,$ OR$,$ NOT, XOR$),$ not notation like $=1$ or $<=31.$ Provide the simplified version of the derived expression.

b$.$ Draw the circuit from part a$.$