Algebra Level 5

How many ordered pairs of real numbers $$(x,y)$$ are there such that

$\begin{cases} x^2 - y^2 & + \frac{\pi x+\phi y}{x^2+y^2} & = \sqrt{2}, \\ 2xy & + \frac{ \phi x-\pi y}{x^2 + y^2} & = 0. \\ \end{cases}$

( $$\phi$$ is the golden ratio $$\frac{1+ \sqrt{5} } { 2}$$. $$\pi$$ is pi, which is approximately $$3.14159$$.)

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