
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
Let \(a, b, c\) be integers. Define \(f(x) = ax^2 + bx + c\). Suppose there exist pairwise distinct integers \(u, v, w\) such that \(f(u) = 0\), \(f(v) = 0\), and \(f(w) = 2\). Find the maximum possible value of the discriminant \(b^2 - 4ac\) of \(f\).

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