You didn't mention a budget, but if you want to shoot 5-600 yards I seriously doubt the cheap S&W, Ruger, etc. models will give you much, if any, joy.
I'm not going to address build-your-own-from-pieces-parts or custom here - I don't know enough about that process. I'm talking buy-it-load-it-run-it factory options here.
If you have a top-rung budget, it's hard to beat
JP Rifles models.
My personal favorite is not a player on the national stage, but is very well known and respected in central North Carolina: the various models from
Barnes Precision Machine. Andrew's rifles aren't cheap. But every bit of the thing other than barrel blanks is machined right there in the BPM factory, down to the pins holding everything together. The blanks - I forget who he sources from - are machined on-site.
I had (since sold for a .223 bolt gun) a Barnes AR in the DMR configuration. It was a solid 1MOA @ 100 yards rifle with now-discontinued Hornady Steel Match 75gr BTHP ammo, reliable 2-2.5MOA rifle 300-500 meters. I wasn't handloading .223 at the time; I'm sure it would have done better with good ammo.
I bought that rifle used. About a hundred rounds in, it began occasionally locking open with one round left in the mag. Short story: I live 20 minutes from BPM's shop, and the company owner greeted me there, fiddled with the mag latch for maybe 10 seconds, and said there was a bit of something in the latch's spring well, showing me how the latch had some slop in it. He disappeared with it for maybe five minutes, gave it back to me, and it never, ever had any kind of malfunction for the couple of thousand rounds I put through it before trading it.
That crew knows ARs - both standard and large frame. They do some interesting things for the military (especially since Fort Bragg is just over an hour away). Any time I had a question - this was my first decent AR ever - I emailed the company "info" account and had an answer back from the boss within a day.
Since I started this, you specified a budget of $1000. You can get a Barnes basic rifle for that amount. The fully-featured ones are double that. I don't know what the new ones actually sell for (as compared to the prices on their web site). Yeah, you can build your own from pieces parts cheaper, at the potential "cost" of a steep learning curve figuring out why you can't get decent groups out of it.
If you want to shoot 400-yard coyotes, my $0.02 on optics would be something in the 3-15x range. I had a 2.5-10x gen-1 Vortex Viper PST on mine, and, while it was adequate (and light), were I to do it again I'd use a 3-15x44 gen-2 Viper PST (which I use on one of my .22s). It's still reasonably light - and Barnes rifles are NOT featherweights anyway.
Oh, and if you buy bulk ammo - the stuff that's advertised "cheap" all the time - forget about MOA with it. It's good for plinking, but not much else. You'll want a 1:7 or 1:8 twist barrel for decent 69-77gr match bullets.
Good luck!
