Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n).$ Please provide the value of $f(n)$ when $n=2$. Directly give the value of $f(n)$. 

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