Integer solution of $2 m (2^{n+1} - 1) = n (3^{m+1} - 1)$

Question

I was reading a lot of other questions here which are supposed to be similar, but none of the answers gave me a hint, how I can approach this. I wrote a Python program to solve it, but up to $2^n < 1e20$ there were no solutions. So I guess my question should be, how do I prove that this equation has no integer solutions other than the trivial solution n = m = 0?