
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

For which values of \( n \) is the polynomial \((x+1)^{n} - x^{n} - 1\) divisible by:
a) \( x^{2} + x + 1 \)
b) \( \left(x^{2} + x + 1\right)^{2} \)
c) \( \left(x^{2} + x + 1\right)^{3} \)?

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