
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 any positive integer \( n \in \mathbb{Z}^{+} \), let \( D_{n} \) be the greatest common divisor of all integers of the form \( a^{n}+(a+1)^{n}+(a+2)^{n} \) where \( a \) is a positive integer. Determine the set \( \left\{ D_{n} \mid n \in \mathbb{Z}^{+} \right\} \).

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