An Archimedean spiral starting at the origin turns counterclockwise and has its first intersection with \(y = 0\) at \(x = -\pi\). The spiral satisfies

\[x^{2}+y^{2} = f \left(\frac{x}{y} \right).\]

Find \(\displaystyle \lim_{z \rightarrow 0} f(z)\).

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