
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 \(F(x)=|f(x) \cdot g(x)|\), where \(f(x)=ax^{2}+bx+c\) and \(g(x)=cx^{2}+bx+a\) for \(x \in [-1,1]\). For any \(a, b, c\) that satisfy \(|f(x)| \leq 1\), determine the maximum value of \(F(x)\) as \(a, b, c\) vary.



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