Callous Constraint

Algebra Level 5

Positive reals \(x\), \(y\), and \(z\) are such that \(\ x^3 + y^3 + z^3 = \left(3 + \dfrac{1}{666}\right)xyz\).

Find the smallest integer value of \(k\) satisfying the following inequality:

\[\large \dfrac{x}{y}+\dfrac{y}{z}+\dfrac{z}{x} \ge 3+\dfrac{1}{k}.\]

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