
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 \( n \in \mathbf{N} \) do we have
(1) \( x^{2}+x+1 \mid (x+1)^{n} - x^{n} - 1 \) ?
(2) \( x^{2}+x+1 \mid (x+1)^{n} + x^{n} + 1 \) ?

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