
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 triples \((a, b, c)\) such that \(a \cdot b + b \cdot c + c \cdot a = 1\) and \(a^2 \cdot b + c = b^2 \cdot c + a = c^2 \cdot a + b\) hold.

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