\[\lim_{n \to \infty} (x^2+y^2)^\frac 1 n =\arctan_{n}(e^{\arctan \left(\frac yx \right)}-1)\]

Find the area of loops of the curve above.

**Notation:** \(\arctan_{n}\) represent a function \(\arctan (\cdot)\) repeated \(n\) times, for example: \(\arctan_3 (x)= \) \(\arctan(\arctan(\arctan(x)))\).

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