I'm not sure if a hint is needed, or if this problem is not challenging at all. I'm leaning towards the latter.
Also as is multi-valued the right hand side of final line is just one of the solutions that correspond to the left hand side (as you know, and I know you know).
As where for we have:
1. this is at least the third refinement of the original idea scribbled in my note book
2. the multi-valuedness of the solution is not only dependent on , but the Lambert-W is itself multi-valued)