- "He works on..." means that he carries out actions on the matters.
- "He deals with..." means that he carries out actions that lead to a resolution of the matters.
- "Work on the problem" means "try a bit harder."
- "Deal with the problem" definitely means "complete the problem."
I can't think of other ways of saying "work on" or "deal with" that sound as elegant. You sometimes come across phrasings like:
- "Action the problem."
- "Progress the problem."
To me, neither of these sound elegant.
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