2013 Mazda3 rough idle

2013 Mazda 3 Touring has been idling somewhat roughly for several months when I leave it on and in Park. I can feel the car's engine vibrate slightly. When I reverse out of a parking space and then switch to Drive and lightly press the gas pedal the car won't move for a couple seconds. Then it will jerk for a moment and then be fine the rest of the drive.

A few months ago the Mazda stopped running when I was out for lunch at work and Pep Boys said there was an issue with a 02 sensor and they cleaned the fuel injectors. They said the battery and alternator were fine. The car still idles somewhat roughly after that. Also, the used car came with an aftermarket remote start which I never use. And a couple months ago I changed the spark plugs which helped a little.

What could be the problem?
 
How many miles are on the odometer? Is your engine a 2.0L or 2.5L?
This may be something a dealership will have to diagnose and resolve.
 
Could be a bad coil pack as well. That's super easy and quick to diagnose and repair if needed. All you need is a pocket screwdriver and a 8mm socket.
 
Wouldn't that be uncommon at 70,000 miles? I guess it's worth a check.
 
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