What should I do?

So there's this guy that I never trusted his actions around my girlfriend , so first I told my girlfriend to not give him much attention and be away from him as much as possible , and I trusted her to do so , but turns out she was casually speaking with him all the same like before ' way too much ' , which kinda broke a bit of my trust , but I let it go by telling her to stop talking to him for good , and that went on for 2 months when I just recently knew that she was talking to him like usual and everything , 3 days ago I asked her if she was talking with him like before , and she lied and said she's not.
What should I do? should I forgive her? Or should I not be with a liar? she already broke a lot of my trust
What should I do?
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