To receive an account verification code, you just need to have two-factor authentication set upon a device or computer that Apple already trusts.
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Re: "I have tried to follow the steps many times without success".
Am unsure what method you have been trying ... as you have established, it doesn't work for you. You may need to try something else.
Eg: Check your account details are correct on your iPhone:
Open Settings > [ your name ] > Sign in and Security
Check your details and phone number are correct and up-to-date
If so, you can use them for verification.
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If you've tried that already:
Try logging in to your account online at https://account.apple.com/
You will likely receive a verification code ok on your iPhone:
How to: Get a verification code and sign in with two-factor authentication - Apple Support
Else: If you forgot the Apple Account password:
Verify your account information to reset your password.
Go to: https://iforgot.apple.com/
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Re: " ... warning that "some data is not synching" ... "
See what to do: If you get an "iCloud Data Not Syncing" message - Apple Support
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If needed:
Apple support website (info. / support / contact options etc.)
Official Apple Support