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Yes, Have you checked it for trouble codes in the on board diagnostic system. That is the best next step from where you are at. Most larger parts stores rent the device you need to read the codes. Odds are it is an electronic problem causing the injectors to not give it enough fuel to run. When it is cold it will inject a little more fuel, that is probably it tries to start in the mornings.
I am checking through a data base I keep at my shop, I am trying to see if there is a common failure for this model making it act like this. I will be right back with you.
What engine do you have