$($a$)$ Using nodes: Assume that the comparator saturates at $9$ V$,$ and that the reference voltage is VREF$=5$ V$.$ If the resistor R$2$ is composed of a fixed resistor in series with a potentiometer, design the circuit such that the zero crossing voltage is variable in a range of $-2$ V to $-4$ V$.$ The minimum resistance that you should use is $20$ k$.$ Additionally, could you explain how the circuit works in detail? What does the parallel R$1$ and R$2$ influence or what is the purpose of placing this parallel of resistors?

$($b$)$ Redesign the circuit so that the zero crossing voltages are $2\mathrm{.}5$ to $3\mathrm{.}5$ V$.$ Explain in detail how you redesigned the circuit.