**valid label** is one where each edge is labeled a distinct number from 1 to 12. An **increasing path** is a path formed by the edges labeled $i, j$ and $k$ such that $i < j < k$ and the edges $i, j$ and $k$ form a continuous curve (i.e. edge $j$ has a common vertex with edges $i$ and $k$, and these 3 edges do not share a common vertex). Over all possible valid labels, what is the minimum number of increasing paths?

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