Solve the system of functional equations.

Question

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \begin{cases} f(x(1+f(x)))=f(x)^2,\\ f(x(1-f(x)))=f(x) f(-x),\\ f(x(-1+f(-x)))=f(x)f(-x). \end{cases}

I have found only that for nontrivial $f$ must be $f(0)=1$ and have no more ideas.