Tri-colouring a Triangular Grid
Consider the following configuration of $64$ triangles:
We wish to colour the interior of each triangle with one of three colours: red, green or blue, so that no two neighbouring triangles have the same colour. Such a colouring shall be called valid. Here, two triangles are said to be neighbouring if they share an edge.
Note: if they only share a vertex, then they are not neighbours.
For example, here is a valid colouring of the above grid:
A colouring $C^\prime$ which is obtained from a colouring $C$ by rotation or reflection is considered distinct from $C$ unless the two are identical.
How many distinct valid colourings are there for the above configuration?