Advanced System Of Equations

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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