Shall I use "not to", "not", or "don't"?

Question

What to use in the following sentence?

You need to believe in yourself and not to/not/don't give up after the first failure.

It seems to me that the correct option is the first one, but I can't explain why and am not even sure if it really should be not to.

Answer 1

The preferred formulation is: "You need to believe in yourself and not give up after the first failure." The formulation with "not to" is also accepted.

Here are a few examples found "in the wild" (emphasis added); the last example uses "not to":

• '"You must believe in yourself and not listen to too much negativity," she says.' (Glamour Magazine)
• "At a time when they could have felt under enormous pressure, he made them believe in themselves, and not be intimidated by the challenge." (Executives Online)
• "The theme of the book is to encourage people to stand tall, believe in themselves and not let people around you control the way you live your life." (Michael Panckridge)
• 'Her inspirational message to “any woman who has ever been put down”, is to believe in themselves and not listen to the naysayer’s. In her message she admits “not listening to any of it”, when she was told discouraging opinions by people close to her.' (Go Ask Mum)
• "We should sponsor entrepreneurship as PayPal co-founder, Peter Thiel, has done with his “20 Under 20” initiative, encouraging smart youngsters to believe in themselves and not to fear failure." (The Telegraph)