
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
Determine all triples of real numbers \((a, b, c)\), with \(a \neq b, a \neq 0, b \neq 0\), such that the parabolas

\[ y = a x^2 + b x + c \quad \text{and} \quad y = b x^2 + c x + a \]

have the same vertex.

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