# To The Sixth

Algebra Level 5

Suppose that the following holds for complex numbers $$a$$ and $$b:$$ \begin{align} \left(a - \frac{1}{b}\right)\left(b - \frac{1}{a}\right) & = a^2 + ab + b^2 \\ \frac{1}{a} + \frac{1}{b} + \frac{1}{ab} & = a + b. \end{align} Then the largest possible real value of $$a^6 + b^6$$ can be expressed in the form $p + \frac{q}{\sqrt{r}},$ where $$p,q,r$$ are positive integers, $$q$$ and $$r$$ are relatively prime, and $$r$$ is square-free.

What is the value of $$p + q + r$$?

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