This is the Great Rhombicosidodecahedron (one of the Archimedean Solids):

In this picture it is colored using \(3\) different colors.

However, suppose you have \(62\) different colors, and you wish to paint each of its \(62\) faces in a different color.

Let \(\dfrac{a}{b}\) be the ratio of the number of **unique** ways to paint this solid divided by the total number of ways this solid can be painted, where \(a\) and \(b\) are coprime positive integers.

What is \(a+b\)?

**Note**: Unique implies that for two coloring combinations \(A\) and \(B\), you can't pick up \(A\) and reorient it to make it look exactly like \(B\).

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