Comparing fractions: GCD

\[ \large \dfrac{\gcd(a+b, b+c, c+a)}{\gcd(a, b, c)}\]

Find the sum of all possible values of the fraction above given that \(a,b\) and \(c\) are any positive integers.

Clarification: \(\gcd(\cdot) \) denotes the greatest common divisor function.

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