This problem is asking for I/o function, i.e V2/V1. I tried to solve this problem by first figuring out the Thevenin equivalents of input voltages and resistance before op-amp's input terminals but this approach is being discouraged by the question. Any insights on solving this problem will be appreciated.
Use the Golden Rules of op amp circuit analysis. Since you have negative feedback, the voltages at the inverting and non-inverting inputs are (effectively) the same. With that information you can figure out how much current is flowing through the right-most 2R resistor. You could use any common circuit analysis method, but if you redraw that part of the circuit an easy approach may become obvious.
The other Golden Rule is that no current flows into the inverting or non-inverting input. So, where does the current flowing through the right-most 2R have to go? You now know the current through the feedback resistor as well as the voltage at one end of it, so finding an expression for the output voltage should be easy.
Rearrange this expression to get V2/V1 on one side. Done!