# Valid Sets (The actual NIMO 2012 3.9)

Let $$f(x) = x^2 - 2x$$. A set of real numbers $$S$$ is valid if it satisfies the following:

1) If $$x \in S$$, then $$f(x) \in S$$.
2) If $$x \in S$$ and $$\underbrace{f(f(\dots f}_{k\ f\text{'s}}(x)\dots )) = x$$ for some integer $$k$$, then $$f(x) = x$$.

Find the number of 8-element valid sets.

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