Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f:\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ is called $k$-good if $\gcd(f(m)+n,f(n)+m)\leqslant k$ for all $m\neq n.$ Find the smallest positive integer $k$ such that there exists a $k$-good function.

Let's think step by step and output the final answer within \boxed{}.
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