A function \(f: \mathbb{N} \rightarrow \mathbb{N}\) is called circular if for each \(p \in \mathbb{N}\) there exists \(n \in \mathbb{N}\) with \(n \leq p\) such that

\[

f^n(p)=\underbrace{f(f(\ldots f}_{p \text { times }}(p)) \ldots)=p .

\]

The function \(f\) has repulse degree \(k, 0<k<1\), if for each \(p \in \mathbb{N}\) we have \(f^{\prime}(p) \neq p\) for all \(0<i \leq[k p]\). Determine the biggest repulse degree that can be reached by a circular function.