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Check-shove the flop. Guy isn't betting 99-QQ if he has them, but he's probably betting air. He'd bet top pair or better too, but he has it well under half the time IMO.
As for what people have when they flat 4-bets. Sometimes it's AA trapping you, but usually when there's history guys will just shove with AA because they know you're calling their shove most of the time, and it'd be a disaster if you had AK missed the flop and check-folded. Usually they have a hand like 99-QQ or some suited broadway hand that didn't believe you preflop and wants to force you to put your stack in, and plans take the pot away if you give up.
I think it's horrid play, but you have to know how to react to it. Their range is weak, so you definitely need to keep the pressure on them and not let them take many pots away cheaply. If you're actually check-folding a lot of flops then they might turn a profit with their flat call.
Here's a fun hand from a couple weeks ago. Flop call was instant. 
PokerStars No-Limit Hold'em, $6 BB (5 handed) Poker-Stars Converter Tool from FlopTurnRiver.com (Format: FlopTurnRiver)
UTG ($622)
Hero ($1367)
Button ($603.50)
SB ($755.65)
BB ($557.70)
Preflop: Hero is MP with 8 , A .
1 fold, Hero raises to $18, Button raises to $60, 2 folds, Hero raises to $150, Button calls $90.
Flop: ($309) 2 , 8 , 9 (2 players)
Hero checks, Button bets $453.5 (All-In), Hero calls $453.50.
Turn: ($1216) 3 (2 players, 1 all-in)
River: ($1216) A (2 players, 1 all-in)
Final Pot: $1216
Results in white below:
Hero has 8c Ac (two pair, aces and eights).
Button doesn't show Qh Jh (high card ace).
Outcome: Hero wins $1216.
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