Q$(4)$ A given processor has $32$ registers, uses $16-$bit immediate numbers and has $142$ instructions in its ISA. In a given program: $20\%$ of the instructions take $1$ input register and have $1$ output register, $30\%$ have $2$ input registers and $1$ output register, $25\%$ have $1$ input register, $1$ output register and take an immediate input as well, and the remaining $25\%$ have one immediate input and $1$ output register. a$)$ For each of the $4$ types of instructions, how many bits are required? Assume that the ISA requires that all instructions be a multiple of $8$ bits in length. b$)$ How much less memory does the program take up if variable$-$length instruction set encoding is used as opposed to fixed$-$length encoding?