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not either
vs
neither

When you agree with someone in the negative you can use not either or neither.

For example B agrees with A in the negative:-

A: I don't like cheese.

If you don't like cheese, you can show your agreement in different ways:-

B: I don't like it either. / Neither do I. / Me neither.

A: I didn't enjoy the film.
B: I didn't like it either. / Neither did I. / Me neither.

And here's one for Joao.

Me: I am never alone.
Joao: I'm never alone, either. / Me neither.

See also either / as well / too
See also either or vs neither nor