Determining gain, KCL/KWL (Basics)
Well, we have the following circuit:







simulate this circuit &ndash; Schematic created using CircuitLab

I used Mathematica to write a code using KCL and KVL. The code I came up with is the following:

FullSimplify[
 Solve[{Iin == I1 + I2, I3 == I2 + β1*I2, 
   0 == I4 + I5 + β1*I2, Ie == I5 + β2*I5, Ie == I6 + I7, 
   I1 == Ia + Iin, Ib == I3 + Ia, 0 == I4 + Ib + Ic, 
   Id == Ic + β2*I5, Id == I6 + I7, I1 == (Vin)/R1, 
   I2 == (Vin - V1)/R2, I3 == (V1)/R3, I4 == (V2)/R4, 
   I5 == (V2 - V3)/R5, I6 == (V3)/R6, I7 == (V3)/R7}, {Iin, I1, I2, 
   I3, I4, I5, I6, I7, Ia, Ib, Ic, Id, Ie, V1, V2, V3}]]


Using your values I got the following code:

FullSimplify[
 Solve[{Iin == I1 + I2, I3 == I2 + β1*I2, 
   0 == I4 + I5 + β1*I2, Ie == I5 + β2*I5, Ie == I6 + I7, 
   I1 == Ia + Iin, Ib == I3 + Ia, 0 == I4 + Ib + Ic, 
   Id == Ic + β2*I5, Id == I6 + I7, I1 == (Vin)/100000, 
   I2 == (Vin - V1)/1530, I3 == (V1)/R3, I4 == (V2)/2200, 
   I5 == (V2 - V3)/359, I6 == (V3)/300, I7 == (V3)/200}, {Iin, I1, I2,
    I3, I4, I5, I6, I7, Ia, Ib, Ic, Id, Ie, V1, V2, V3}]]


I left \$\text{R}_3\$ as it was because I need to set it equal to \$0\$ and than I will get an error message because then I will divide by \$0\$. Later I will take the limit.

Entering the code I got the following result:

{{Iin -&gt; Vin (1/100000 + 1/(1530 + R3 + R3 β1)), 
  I1 -&gt; Vin/100000, I2 -&gt; Vin/(1530 + R3 + R3 β1), 
  I3 -&gt; (Vin (1 + β1))/(1530 + R3 + R3 β1), 
  I4 -&gt; -((Vin β1 (479 + 120 β2))/(
    3 (1530 + R3 + R3 β1) (893 + 40 β2))), 
  I5 -&gt; -((2200 Vin β1)/(
    3 (1530 + R3 + R3 β1) (893 + 40 β2))), 
  I6 -&gt; -((880 Vin β1 (1 + β2))/(
    3 (1530 + R3 + R3 β1) (893 + 40 β2))), 
  I7 -&gt; -((440 Vin β1 (1 + β2))/((1530 + R3 + 
       R3 β1) (893 + 40 β2))), 
  Ia -&gt; -(Vin/(1530 + R3 + R3 β1)), 
  Ib -&gt; (Vin β1)/(1530 + R3 + R3 β1), 
  Ic -&gt; -((2200 Vin β1)/(
    3 (1530 + R3 + R3 β1) (893 + 40 β2))), 
  Id -&gt; -((2200 Vin β1 (1 + β2))/(
    3 (1530 + R3 + R3 β1) (893 + 40 β2))), 
  Ie -&gt; -((2200 Vin β1 (1 + β2))/(
    3 (1530 + R3 + R3 β1) (893 + 40 β2))), 
  V1 -&gt; (R3 Vin (1 + β1))/(1530 + R3 + R3 β1), 
  V2 -&gt; -((2200 Vin β1 (479 + 120 β2))/(
    3 (1530 + R3 + R3 β1) (893 + 40 β2))), 
  V3 -&gt; -((88000 Vin β1 (1 + β2))/((1530 + R3 + 
       R3 β1) (893 + 40 β2)))}}


Now using your gain formula we will end up with:

$$\text{G}:=\frac{\text{V}_3}{\text{V}_\text{in}}=\lim_{\text{R}_3\to0}-\frac{88000\beta_1\left(1+\beta_2\right)}{\left(1530+\text{R}_3\left(1+\beta_1\right)\right)\left(893+40\beta_2\right)}=$$
$$-\frac{8800\beta_1\left(1+\beta_2\right)}{153\left(893+40\beta_2\right)}\tag1$$