\[ \begin{cases} F^2+A^2+N^2 =3 \\ \min \left( F+A, A+N, F+N \right) > \sqrt 2 \\ \frac{F}{\left( A+N-F \right)^2}+\frac{A}{\left( N+F-A \right)^2}+\frac{N}{\left(F+A-N \right)^2} \geq \frac{S}{3000 \left( FAN \right)^2} \end{cases} \]

For what maximum value of \(S\) will the above conditions be always met simultaneously for positive real numbers \(F,A\) and \(N\)?

**Clarification:**

- \(FAN= F \times A \times N\).

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