Tricky Sum

Algebra Level 3

\[ \large \dfrac{ \displaystyle abc + \left( \sum_{\text{cyclic}} a \right) + 6 }{\displaystyle \left( \sum_{\text{cyclic}} a \right) + 8} = \dfrac2{abc} \]

If \(a,b\) and \(c\) are positive integers satisfying the equation above, find the value of \(a+b+c\).

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