Radically Remarkable Root

Algebra Level 5

6x33x26x4=0\large 6x^3-3x^2-6x-4=0

The real root to the equation above can be written in the form a3+b3+1c\large \dfrac{\sqrt[3]{a}+\sqrt[3]{b}+1}{c}, where aa, bb and cc are positive integers.

Find a+b+ca+b+c.

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