62 Colors

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 \(B\).

Image credit: commons.wikimedia.org

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