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# Advanced Factorization

Advanced factorization is a gateway to algebraic number theory, which mathematicians study in order to solve famous conjectures like Fermat's Last Theorem.

If \(P(n)=\frac{n^3+n^2-2n}{n^2-4}+\frac{n+1}{2+n-n^2}\), what is the value of \(\lfloor P(409) \rfloor + \lfloor P(423) \rfloor\)?

**Details and assumptions**

The function \(\lfloor x \rfloor: \mathbb{R} \rightarrow \mathbb{Z}\) refers to the greatest integer smaller than or equal to \(x\). For example, \(\lfloor 2.3 \rfloor = 2\) and \(\lfloor -5 \rfloor = -5\).

\(x, y\) and \(z\) are complex numbers such that

\[ \begin{cases} x + y + z & = 46 \\ (x-y)^2 + (y-z)^2 + (z-x) ^ 2 & = xyz.\\ \end{cases} \]

What is the value of \( \frac{ x^3 + y^3 + z^3} { xyz} \)?

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