Not another composed function

Algebra Level 4

Define the function f(x)=2x1x2f(x)=\frac{2x}{1-x^2}. Find the number of distinct real solutions of the equation f(5)(x)=x.f^{(5)} (x) =x.

Details and assumptions

f(n)(x) f^{(n)} (x) denotes the function ff applied nn times. In particular, f(5)(x)=f(f(f(f(f(x))))) f^{(5)} (x) = f(f(f(f(f(x))))).

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