
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
Let  $X=\mathbb{Z}^3$  denote the set of all triples  $(a,b,c)$  of integers. Define  $f: X \to X$  by \[ f(a,b,c) = (a+b+c, ab+bc+ca, abc) .  \]
Find all triples  $(a,b,c)$   such that \[ f(f(a,b,c)) = (a,b,c) . \]

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