
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 real numbers \((a, b, c)\) that satisfy the following conditions:
\[
\begin{array}{l}
a^{2}b + c = b^{2}c + a = c^{2}a + b, \\
ab + bc + ca = 1.
\end{array}
\]

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