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Unfortunately, they were not a legitmate company and what you have experienced was a tech support scam. However, the good news is that most banks, merchants and credit card providers are aware of such tech support scams so users who've been scammed can initiate a charge back with their banks or with whatever payment method they have used to get the money back. Let me know if you have any questions.
It depends on what type of access they had, if they told you to download number and password, a program and give them a 9 digit number and a password (TeamViewer) then they can if they have the password, if they used LogMeIn and gave you a six digit code, then they can't cause they will need you to enter a new code. The program they have used, if it's there - should be uninstalled.
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