Find the cardinality of the set of all arithmetic functions satisfying all of the following properties:

(1) For any $n\in \mathbb{N}$, $f(n)^{n}=1$.

(2) For any $n\in \mathbb{N}$, $f(n)^k \neq 1$ if $0< k < n$.

(3) If $m|n$, then $f(n)^{\frac nm}=f(m)$.

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