I am having trouble with the following circuit. This circuit is AC model of Colpitts oscillator designed with NPN type BJT transistor. I am trying to find loop gain and check if it meets Barkhausen criteria. I wrote down bunch of node analysis equations, but I cannot eleminate intermediate voltages such as Emitter voltage or Base voltage. I am totally stuck. I even tried to use MATLAB and its symbolic toolbox to eleminate intermediate voltages and find gain of this circuit, which failed terribly. In this circuit below Rin is output voltage of the DC source. If it is not possible to solve this circuit with all the components already present, Rin, Cin Ro and internal capacitances of BJT can be ignored to simplify it. Operating frequency is approximately 1Mhz and I naively believe that this circuit is not classified as high frequency circuit, therefore internal capacitances can be seen as open circuits, but I am not pretty sure. Any help or advice that could help me solve this circuit is highly appreciated.
By the way RE2 is advised to have as it is said that it provides negative feedback, therefore makes output voltage not distorted.