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In the meantime: do not change your password, email address or Twitter handle. Until Twitter corrects the issue, it's best not to attempt any of these changes at this time.
They go on to say that if you have made any of these changes but do manage to successfully log in, try to stay logged in for a few hours and don't use any third party applications. They suspect it may be a caching issue, and have prioritized it to be fixed as soon as possible.