2001 ford Taurus wont start when its cold out?

Hi, I have a car that Im having trouble with. For awhile I could get it started in the cold weather just by hooking up a charger and it would start right away. The battery is new and at full. We think its a fuel problem, however the heater in this car doesn't work either. Any suggestions, where do I start with my trouble shooting?
when the weather is warm, the car starts every time.
Added (1). I am sorry, yes it does turn over.

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