An algebra problem by madhav srirangan

Algebra Level 5

\[ 729x^{6}-1458x^{5}-1215x^{4}-540x^{3}-135x^{2}-18x-1=0\]

The above equation has a positive real root in the form

\[ \frac13(a^{1/6}+b^{1/6}+c^{1/6}+d^{1/6}+e^{1/6}+f^{1/6}). \]

with \(a, b, c, d, e, f\) are all integers. Find \(a+b+c+d+e+f+3\).

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