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Algebra Difficulty 5.0 AIME, harder Find the answer Hungary

Find all functions $f\colon \mathbb{Z}^+\to \mathbb{R}^+$ that satisfy $f(nk^2)=f(n)f^2(k)$ for all positive integers $n$ and $k$ , furthermore $\$\lim\limits_{n\to \infty} \frac{f(n+1)}{f(n)}=1\$.$

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