
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
Every card in a deck has a picture of one geometric object - cube, sphere, or cylinder, which is painted in one of the three colors - red, blue, or green. Furthermore, each color is applied in one of three shades - light, medium, or dark, and each card is available in three sizes - small, medium, or large. The deck has 81 cards, with every object-color-shade-size combination represented. A set of three cards from the deck is called complementary if all of the following statements are true:
i. Either each of the three cards has a different object or all three of the cards have the same object.
ii. Either each of the three cards has a different color or all three of the cards have the same color.
iii. Either each of the three cards has a different shade or all three of the cards have the same shade.
iv. Either each of the three cards has a different size or all three of the cards have the same size.
How many different complementary three-card sets are there?

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