How does getting a credit card raise credit score?

I will be needing a car loan before the end of the year. My score is 639. I have no mortgages and no outstanding loans or debt. I have never owned a credit card. It is frustrating that having no debt or current loans negatively affects my score. It s like being punished for being financially responsible. How fast and how much would getting a credit card and paying it off every month improve my score? I need to find a way to improve my score in a short amount of time.

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