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

**Details and assumptions**

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

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