
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
Given the sequence \(\{a_n\}\):
\[ a_n = 2^n + 3^n + 6^n + 1 \text{ for } n \in \mathbb{Z}_+ \]

Is there an integer \( k \geq 2 \) such that \( k \) is coprime with all numbers in the sequence \(\{a_n\}\)? If it exists, find the smallest integer \( k \); if it does not exist, explain why.

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