Four (50 points) Let $k \in N^{*}$, if it is possible to color all positive integers with $k$ colors, and there exists a function $f: Z^{+} \rightarrow Z^{+}$ satisfying:
(1) For any same-colored positive integers $m, n$ (which can be the same), $f(m+n)=f(m)+f(n)$;
(2) There exist $m, n \in N^{*}$ (which can be the same) such that $f(m+n) \neq f(m)+f(n)$. Find the minimum value of $k$.