
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
Example 22 (37th Canadian Mathematical Olympiad): An ordered triplet of positive integers $\{a, b, c\}$ is said to be $n$-energetic if it satisfies the conditions $a \leq b \leq c, \gcd(a, b, c)=1$, and $a^{n}+b^{n}+c^{n}$ is divisible by $a+b+c$. For instance, $\{1, 2, 2\}$ is 5-energetic.

1. Find all ordered triplets of positive integers that are $n$-energetic for every $n \geq 1$.
2. Find all ordered triplets that are 2004-energetic, 2005-energetic, and not 2007-energetic.

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