Functional Equation Mood!

It is easy to see that $\phi^n(x)=\frac{x \cos \left(\frac{2 \pi n}{5}\right)+\sin \left(\frac{2 \pi n}{5}\right)}{\cos \left(\frac{2 \pi n}{5}\right)-x \sin \left(\frac{2 \pi n}{5}\right)},$ where $\phi^n$ represents the composition of $\phi$ with itself $n$ times. Therefore, $\phi^5(x)=x.$ Then the following equations hold for any $x$ in the largest set of numbers where the functions $\phi, \phi^2, \phi^3,\phi^4$ are defined, $f(x)+f(\phi(x))=x$ $f(\phi(x))+f(\phi^2(x))=\phi(x)$ $f(\phi^2(x))+f(\phi^3(x))=\phi^2(x)$ $f(\phi^3(x))+f(\phi^4(x))=\phi^3(x)$ $f(\phi^4(x))+f(x)=\phi^4(x).$ Multiplying the second and fourth equations by negative one and adding all of them, we obtain that the only solution $f(x)$ of the given functional equation must satisfy that $2f(x)=x-\phi(x)+\phi^2(x)-\phi^3(x)+\phi^4(x).$ Evaluating at $0$ $2f(0)= -\phi(0) +\phi^2(0)-\phi^3(0)+\phi^4(0)= - \tan \frac{2\pi}{5}+\tan \frac{4\pi}{5}-\tan \frac{6\pi}{5}+\tan \frac{8\pi}{5}= -7.608...$ Then the answer will be $\boxed{760.}$