
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 integer triples \((a, b, c)\) \((a \neq 0)\), such that the quadratic function \(f(x) = ax^2 + bx + c\) has the following property: For any integer \(n\), there exists an integer \(m\) that satisfies
\[ f(m) = f(n) f(n+1). \]

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