Need Help Selecting New Projector

Hello! I live in canada, and I have been looking for a projector for my basement for quite some time now. The previous owner did have one, but it is quite old and out of quality now. I want to have one that's great for movies and gaming, as my family would like this too. It would be around 8 ft to 12 ft away from the wall. I tried searching online, but too many choices and no info sort of overloaded me, so here I am now lol. I really would like this to be good for gaming, with a low input lag, and great for movies too, something that is watchable. Basically, I just want something great for movies and gaming. I do have a budget of $500 CAD, if that does help. Thank you all!