It would be best if you could provide the actual patent number. Many people confuse patent applications from actual patents. Applications may or may not get issued, and the claims are almost always narrowed in an issued patent from its application. Also, if we know the patent number, we can help identify equivalent foreign patents.
I'll try to answer your questions one at a time:
- produce and sell the tool in a country not covered by the patent
This is certainly an option, but you should be advised that companies often patent the same invention in multiple countries, so you need to look for those. Of course this option limits your potential market.
- approach the company and propose a licensing agreement
This is typically the best option if you want to avoid getting sued. You will of course reduce your profitability by the licensing terms. There is every possibility that the patent owner may decline to license the patent.
- improve upon the claims covered by their patent
Although you may be able to get a patent on an improvement to an existing patent, this doesn't, by itself, get around infringing on the original patent. If you implement every step of even a single claim of an existing patent, you infringe on it. An improvement patent can sometimes be used to improve your licensing terms by virtue of a cross license.
- find some innovation not covered by their claims
If you can avoid infringement then that is great and you avoid paying licensing fees. You can potentially patent the innovation. You always have to be aware that there is potential for other patents that cover the "other innovation".
- Am I correct that those are my options?
- Are there any other options I may not be aware of?
I think you covered the options pretty well. Perhaps the only other option would be to find some basis upon which to invalidate the existing patent. This, I believe, is not a simple process for existing patents, but if the document is actually an application and you can identify some relevant prior art, this might be an option.
Please be advised that I am not a patent lawyer and this is not legal advice. The best course of action is probably to consult with an actual patent attorney to evaluate whether your idea actually infringes this or other patents. These are called "freedom to operate opinions". Many people misinterpret the breadth of specific patents because the title and body of the patent can sound quite general. However, the claims specify what is covered. For instance a claim might state you have a process implementing steps A, B, C and D. If you can implement your process with steps A, B, C but not D, you don't infringe. If you can implement your process with steps A, B, C and E where E and D are not equivalent, you still probably avoid infringement. Consulting with a patent attorney is the best way to evaluate claim language.