Song's Complex System

Algebra Level 5

Let $$a, b, c$$ be complex numbers satisfying

$a + b + c = abc = 1$

and

$\frac{ab + bc + ac}{3} = \frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}}$

The sum of absolute values of all possible $$ab + bc + ac$$ can be written as $$\frac{\sqrt{n}}{m}$$, where $$n$$ and $$m$$ are positive coprime integers. What is $$n+m$$?

This problem is posed by Zi Song Y.

Details and assumptions

You may read up on Absolute Value.

$$n$$ is allowed to be 1.

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