Is This Even Possible?

For how many natural numbers \( n < 1000 \) are there, such that there exists a multiset of \(n\) integers such that sum of \(n\) of them is \(0\), and the product of elements of that set is equal to \(n\)?

A multiset is a collection of objects where order doesn't matter (unlike a sequence) but multiplicities do matter (unlike a set). For example, \(\{1,1,2,3\} = \{2,1,3,1\} \neq \{1,2,3\}\).

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