Daniel's path sums

For integers \(n > 2\), let \(G_n\) be a complete graph on \(n\) vertices such that each vertex is labeled by a distinct number \(1,2,3,\cdots,n\), and each edge is labeled by the sum of its endpoint labels. Let \(f(G_n)\) be the minimum sum of edge labels in any path that touches every vertex in \(G_n\) exactly once.

How many values of \(n\) satisfy \[f(G_n)\equiv 2013\pmod{n}?\]

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