
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
Find all ordered triples of positive integers \(\{a, b, c\}\) such that:
1. \(a \leq b \leq c\) and \(\gcd(a, b, c) = 1\);
2. \(a + b + c\) divides \(a^{12} + b^{12} + c^{12}\), \(a^{23} + b^{23} + c^{23}\), and \(a^{11004} + b^{11004} + c^{11004}\) simultaneously.

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