Factorization is an art

Algebra Level 4

A root of \(10x^3-3x^2-3x-1=0\) is of the form \(\dfrac{\sqrt[3]{a}+\sqrt[3]{b}+1}{c}\), where \(a\), \(b\), and \(c\) are positive integers. Find \(a+b+c\).

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