It is given that \(a\), \(b\), and \(c\) are nonzero real numbers such that

\[(a+b)^2 = (b+c)^2 = (c+a)^2\]

Find the product of all possible distinct values of \(\frac{a}{b} + \frac{b}{c} + \frac{c}{a}\).

If you think that the equality has no solution, submit 0 as your answer.

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