Something interesting in cubes

Algebra Level 4

\[\large 8x^{3}-12x^{2}-6x-1=0\]

If the value of a real root that satisfy the equation above can be expressed as

\[\large \dfrac{\sqrt[3]{a}+\sqrt[3]{b}+1}{c}\]

where \(a,b\) and \(c\) are positive integers, find the value of \(a+b+c\).

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