Double Equality

Algebra Level 5

Suppose that the following holds for complex numbers \(a\), \(b\), and \(c:\) \[\begin{align} \frac{1}{a} + \frac{1}{b} + \frac{1}{c} & = \frac{1}{ab} + \frac{1}{ac} + \frac{1}{bc} \\\\ (a + b)^2 + (a + c)^2 + (b + c)^2 & = (a + b - c)^2 + (b + c - a)^2 + (c + a - b)^2. \end{align}\]

What is the sum of all possible values of \(a^2 + b^2 + c^2?\)

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