A processor from Fasto Technologies $($FT$)$ has $32$ registers, uses $16-$bit immediates 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,

The remaining $25\%$ have one immediate input and $1$ output register.

$(1)$ 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.

$(2)$ How much less memory does the program take up if variable$-$length instruction set encoding is

used as opposed to fixed$-$length encoding?