A method claim in an issued patent is structured as a series of steps or actions. An infringement analysis looks to see if your product takes each and every one of those steps - on a per claim basis. The doctrine of equivalents is not given much use/weight these days but if it was used in an infringement analysis it would be a comparison of one or more particular claimed steps and corresponding ones of your steps to see if they were equivalent. Is your step 2 equivalent to the claimed step 2, for example? It is not a broad comparison of achieving the same result. Patents do not cover results, they cover specific ways of getting a result.
You getting a patent is unrelated to your product infringing or not infringing someone else's patent. There is no such thing as a provisional patent, only a provisional patent application. And the date you get is only as good as the content of the provisional application justifies.