# Possible Increasing Paths in Cubes

Given a cube, a 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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