Tricky Sum

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

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