
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) For which \( n \in \mathbf{N} \), does \( x^{2}+x+1 \mid x^{2n}+x^{n}+1 \) hold?

(b) For which \( n \in \mathbf{N} \), does \( 37 \mid 1 \underbrace{0 \cdots 01} \underbrace{0 \cdots 01} \) hold?

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