If you can't connect to the App Store, iTunes Store, or other Apple services https://support.apple.com/en-us/HT201400. There have been quite a few Apple server connection problems lately. Among other things, in the above document, see Check Apple’s System Status page. You can check there to see if the needed server is available when you need to use one.
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Note also that receiving the message you received is common. But, I have not seen a definitive solution for it. I suggest you use my search link in trying to solve your problem: verification failed connect apple id server. You will see a number of links that are, or may be, related to your specific problem.
And, if you want to contact Apple Support to deal with your problem, see:
Get a call from Apple support to help solve your problem using this
link https://getsupport.apple.com/?caller=kbase&PGF=PGF63005&category_id=SC0245&symptom_id=23362
Apple ID Support https://support.apple.com/apple-id
Apple Support Twitter https://twitter.com/AppleSupport
Apple ID account security for country-specific and
region-specific help https://support.apple.com/en-us/HT204169
Get Support: We’re here to help https://getsupport.apple.com/. Go through
the options to find the best one.