
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
It is known that both KROG and KUB are cubes of certain numbers. Find all possible values of the numbers KROG and KUB. List all of them and explain why there are no others. (Identical digits are replaced with identical letters, different digits with different letters.)

Note: 8 and 1000 are cubes, as $8=2 \cdot 2 \cdot 2=2^{3}$, and $1000=10 \cdot 10 \cdot 10=10^{3}$.

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