Ethics questions concerning a referee assignment I recently refereed a paper that I returned to the author(s) for revision. The thrust of their argument relied on a claim whose justification I felt was lacking. I dutifully raised the issue in my report and, in addition, I corrected another portion of their proof.
The author(s) have yet to revise their work and, in the interim, I came up with a justification for their claim. I now have a proof of this result (which is important for another paper I'm working on) and would like to publish it.

What are the ethics/options here? Do I need to provide them with the correct proof? May I submit the result as my own after their revision? Should I recuse myself from serving as the referee? 

EDIT: Not sure if this makes a difference, but the conjecture the author(s) purports to resolve is one that I raised in a previous a paper and one for which I obtained partial results with similar methods. Although, I have worked on the problem and obtained partial results, I would not have obtained a proof without refereeing their work.
 A: The answer to your question really depends on how much work required justifying their claim. In my personal experience:


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*It was many times that a referee provided me with an argument that lead to a simplification of my poof or even provided me with new results that I included in my paper.

*I did the same many times when I refereed the papers.


Therefore if the justification did not require much work, I would give it to the authors of the paper and let them publish it. Then you would be mentioned as an anonymous referee. 
May I submit the result as my own? If so, must I wait until their revision before I submit my own paper?
That I do not really understand. Since they do not have justification of the claim, they cannot publish it. If they come up with a justification, your proof is of not much value since they already have a proof. In my opinion, you must not publish your proof before their paper is published.
If proving the claim is really difficult and required a lot of work, I do not really know what to advise. In one related situation I did as follows:
I was a referee of a paper $X$. However, before refereeing the paper I already knew how to prove a much better result (5 pages of elementary calculations vs 40 pages of a difficult proofs of far better results). But I only had my proof in a draft. I wrote to the author that I was a referee and I suggested that he would withdraw the paper from the journal and we would publish a joint paper. This is exactly what happened. 
