The average processing time of a machine in a factory is $4$ minutes. The parts planned to be produced arrive at the machine every $5$ minutes on average. The machine can process $4$ different parts sequentially. If the machine is full, the parts are sent to another machine. According to this,

a$)$ On average, how many minutes does the machine remain idle in a $1-$hour period?

b$)$ What is the average number of parts? $()$

c$)$ What is the average number of parts returned without service?

d$)$ What is the average time spent by a part in the machine in minutes?