
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]


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In the sequence of numbers \(100^{100}, 101^{101}, 102^{102}, \ldots, 876^{876}\) (i.e., numbers of the form \(n^n\) for natural \(n\) from 100 to 876), how many of the listed numbers are perfect cubes? (A perfect cube is defined as the cube of an integer.)

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