
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
A cube-shaped cake has all its surfaces (including the bottom) covered in chocolate. We want to distribute the cake among $K$ people so that everyone gets the same amount of cake and the same amount of chocolate coating. This is to be achieved by cutting the cake into $N \times N \times N$ identical small cubes, and giving each person the same number of pieces, ensuring that the total chocolate-covered area of the given parts is also equal. Can this be done for any $K$? At least how many pieces should the cake be cut into if $K = 1997$?

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