What's the best strategy to promote your product and gain engagement on Twitter? Any recommendation?

I think the best way is to find an ideal audience for your product first!

But checking the profiles of all users on Twitter spends a lot of time and effort! So just download hummin(Chrome extension for Twitter) and determine!

Secondly, when you followed many users, segment them: career, interest, just friends or if they could be your customer! Add a tag on the profile and collect people like the tag. It makes you find it easier and more streamlined to contact them later! Because when you tag users, you can see them on your right sidebar without moving the page! And you can directly send DM! Lastly, reply and likes to their tweets. Focus on building a close relationship. Care what they do or what they're into :)