Please reason step by step, and put your final answer within \boxed{}.
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
\[ f(m) = f(n) f(n+1). \]
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