There isn’t one at that price, you will need to choose which is your most usage, if films are a priority than a home cinema projector is what to go for, but game play (Due to home cinema projectors focussing on most films 24p) will be substandard due to lag, motion & dullness, if gaming is your priority then go for a home entertainment projector as these focus on brightness, motion and low lag but which also means they are not so good with films.
Another thing to consider is where is it going, if fixed, the home cinema projectors will be fine, however if you wish to move it around (Or even take it to your friends) then most of them are really too big and heavy, whereas most home entertainment projectors are more compact and lighter and thus more flexible.
If the lead to the projector is longer than about 8m then you will need a powered fibre optic cable for 4K so you will need to allow for this, also the screen very important (You can use a wall but not ideal) so get one to match the light in the room and projector.
Once you have narrowed all the above down, go to a dealer and view the ones you have chosen as there is no one size fits all. (Don’t rely on reviews or other people’s opinions (As its unlikely they will exactly match your requirements) except to narrow down the choice)
Bill