Please answer for the following OP$-$AMP circuit.

$($a$)$ Find $v_0$ when the open$-$loop OP$-$AMP has the input$-$output relationship $v_0=A_d(v^{+}-v^{-}).($Assume $=\frac{R_1}{R_1+R_2}$ and the input current $i^{-}=i^{+}=0.$ Do not use the practical assumption $v^{+}=v^{-}.)$

$($b$)v_1=v_c-\frac{v_d}{2}$ and $v_2=v_c+\frac{v_d}{2}$ when differential voltage and common mode voltages is $v_d=v_2-v_1$ and $v_c=\frac{1}{2}(v_1+v_2)$ respectively. Find $G_d$ and $G_c$ when the output voltage $v_0$ is defined as $v_0=G_d{v}_d+G_c{v}_c.$

$($c$)$ Find the condition of resistances $R_1,R_2,R_3,R_4$ that makes $G_c=0$ and find the output $v_0$ at this condition.