
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 pieces of transparent glass, all of which are identical in square shape and size. Each piece of glass is divided into 4 equal parts (right triangles) by its diagonals, and one of these triangles is painted with an opaque color unique to each glass piece, different from the others. Then, all these glass pieces are stacked on top of each other (with exact alignment of edges and vertices), with the painted parts facing up. How many different ways are there to stack the glass pieces so that the entire stack ends up completely opaque in the vertical direction?

(12 points)

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