Hello IMOment

We say that a set of positive integers \(S\) is nice if it is a non-empty subsets of \(\left\{ 1,2,3,\ldots,2016 \right\}\) and the product of numbers in \(S\) is a perfect power of \(10\).

For example, \(\{2,5,10\}\) is a nice set because of \(2\cdot 5\cdot 10 = 10^2\).

What is the size of the largest nice set?

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