Gaming laptop under $800

Hello everyone, please excuse me for starting a new thread, but I am truly in need of some guidance and haven't been able to find any recent discussions covering this exact topic...
So, here is the situation: in about two months, I’m planning to pick up a laptop specifically for "gaming." I put that in quotes because I’m not looking to become a pro esports player or anything extreme; my main focus would just be playing League of Legends and Grand Theft Auto V (ideally on the highest settings possible 😁).
Now, I won't be running games 24/7. My schedule is a bit unconventional—I might jump on for a bit around 3:00 AM, then maybe a little after 1:00 PM, and finally an hour or two in the evening. I know that sounds a little eccentric, but that's my routine. Regarding the budget, I was thinking about putting about $1667 (MAX $550) toward this. I am open to both brand-new machines or even a used one—if I can find a high-quality, relatively recent model for that price, why wouldn't I? If I go the route of a brand-new laptop, I realize I shouldn't expect much more than an i5 processor paired with a 940m graphics card (I've heard that combo is decent for gaming). My big question is: can an i5 and a 940m actually handle those specific games on medium to high settings? Also, I am currently a sophomore in high school, and I really want this machine to last me at least another 3 or 4 years. Thank you all in advance! 🙂
P.S. If anyone could share some helpful links (including listings from eBay or Craigslist 🙂), I would be extremely grateful.

EDIT: Is it actually possible to find something better than this http://www.bestbuy.com/not...52618/product/

Right now, you can actually snag some seriously decent rigs for that kind of money—machines that’ll handle both heavy work and gaming without breaking a sweat. For example:

Where on earth were you even looking for laptops? Usually, people just hit up Best Buy or Newegg to see what's out there in the US. Besides, you’ve got us to help you out. 😬

Two months is a long time in this industry; tech specs and prices shift constantly. Just come back when you're actually ready to pull the trigger. We keep an eye on the market, so we'll find you something.

And honestly, it wouldn't hurt if you actually read the stickied threads. There's one specifically for laptop buyers and another about gaming on notebooks. There's some useful stuff in there—like how gaming performance really boils down to your GPU and resolution. You can even check Notebookcheck.net to see how specific graphics cards perform in different games; those guys track FPS for everything 😬 and so on...

Tiger, I really appreciate the help... I'll be sure to reach back out in about two months once I'm actually ready to pull the trigger on the purchase 🙂