Based on the circuit shown below, please solve the following three sub parts of this question.

a$.$ Derive the expression for the voltage gain $A_v$ of this circuit; considering both channel length modulation and body effect.

$\{10$ marks $\}$

b$.$ Assuming the sensed input voltage $V_{in}=0\mathrm{.}9V,V_{DD}=1\mathrm{.}8V,$ bias voltage $V_{bl}=1\mathrm{.}2V$; determine the range of output voltage $V_{\text{o}\text{u}\text{t}}$ for which all the transistors in the circuit remains in saturation operating region. Consider $V_{\text{T}\text{H}\text{I}}=0\mathrm{.}5V,|V_{TH}|=|V_{TH2}|=0\mathrm{.}6V$ and $V_{TH3}=0\mathrm{.}7V.$

$\{10$ marks$\}$

c$.$ For this question, consider all the values of voltages in the above part of this question $($b$).$ Now assuming the bias currents $I_{D3}=0\mathrm{.}2mA$ and $I_{DI}=0\mathrm{.}7mA,$ compute the output impedance and voltage gain of this circuit for the $V_{\text{o}\text{u}\text{t}}=1\mathrm{.}7V$ and node voltage at $x$ to be $V_x=0\mathrm{.}8V.$ Let the circuit experience negligible body effect and substantial channel length modulation.

$\{15$ marks$\}$