A PIC$24$H processor with Fosc $=40MHz$ consumes $40$ mA during normal operation, $10$ mA at the idle mode, and $30$ nA at the sleep mode. It is required that the processor runs with the following two conditions: $(1)$ Maximum possible speed, $(2)$ Power consumption is less than $18$ mA $.$

$($a$)$ Suggest a doze mode $($doze$/$N$)$ value.

$($b$)$ How many single cycle instructions can be executed in one second.

$($c$)$ If the processor spends $90\%$ of its time in sleep mode, $9\%$ in doze mode and $1\%$ in the normal mode, what is the average current consumption?

$($d$)$ Based on your answer in $($c$),$ estimate how long a full $500mA.h$ battery operates the processor before the battery becomes empty?