Determine all pairs $(a, b) \in \mathbb{C} \times \mathbb{C}$ satisfying
$$
|a|=|b|=1 \quad \text { and } \quad a+b+a \bar{b} \in \mathbb{R}.
$$
(proposed by Mike Daas, Universiteit Leiden)
Please provide the sum of all possible values of a and b.

Let's think step by step and output the final answer within \boxed{}.
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