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Given the symptoms you describe, I strongly suspect you have an ECT sensor problem - This is the Engine Coolant Temp Sensor - not the one the controls the gauge per se- but the one that sends signals to the cars Ecu.. main computer - re. the temperature of the engine.
The Ecu then sends commands to the fueling and ignition systems according. If the sensor is faulty and thus sends incorrect , or no info to the ecu as above, the fuel and air mixture will be incorrect and thus do as you describe resulting in poor running and no start situations...
If you remember the old auto and manual choke systems...it works like that , only electronically. If and when the choke was in the wrong position for the temp of the engine, the engine either wouldn't start or coughed and spluttered... exactly the same, temp related too..
So, without the aid of plug in diagnostics at the moment and the results thereof - its well worth renewing that sensor. Its usually located on the inlet thermostat housing , inlet manifold or thereabouts depending on model and has a two wires pin connector going to it.
Always buy one from a dealer / genuine unit. If you are unsure as to its position - ask the parts dept; when you are in there, to show you on the screen where it is located ....
That sensor is a common fault and will cause the problems you describe - its cheap and easy to replace as well - so all things considered at this point...I would suggest renew it and see if that cures this for you.
The only other thing to consider here would be if the throttle body were gumming up inside - you haven't mentioned the mileage but if its any where above average - its worth considering giving that a thorough clean and using some fuel system cleaner via the tank.
thing that can often cause such issues a problem with the software in the ecu - the engine management computer. It would be worth asking the dealer if there are any updates for yours - often there are - and often they dont think about this aspect unless prompted.
So there's three considerations - any and all of which will cause what you have.Let me know if you need anything more on this at all - >> >> Meantime -- Please click on the OK rating to Excellent ratings service for me now, Thank you for that... and you can get back to me anytime if you need / wish..
Best Rgds, MIKE. Please note, you are rating the content of my "answer" - my help to you - not the "system" which can experience delays etc which are not under my personal control -or delays caused by "time -off" which we all have on occasion - Thank you.