Let $f$ be a function taking the positive integers to the positive integers, such that
\[f(mf(n)) = nf(m)\]for all positive integers $m$ and $n.$  Find the smallest possible value of $f(2007).$
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Let $f$ be a function taking the positive integers to the positive integers, such that
\[f(mf(n)) = nf(m)\]for all positive integers $m$ and $n.$  Find the smallest possible value of $f(2007).$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
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Let $f$ be a function taking the positive integers to the positive integers, such that
\[f(mf(n)) = nf(m)\]for all positive integers $m$ and $n.$  Find the smallest possible value of $f(2007).$
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