
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
There are 5 square pieces of transparent glass, all of the same size. Each piece is conditionally divided into 4 equal parts (right-angled triangles) by its diagonals, and one of these triangles is painted with an opaque paint in a unique color different from the colors painted on the other pieces of glass. All these pieces are stacked on top of each other (with exact alignment of the edges and vertices) with the painted parts facing up. How many different ways can the glasses be stacked so that the entire stack is completely opaque in the vertical direction?

Present the answer in LaTex format: \boxed{Your answer}
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