
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 \( f(x) = a x^{2} + b x + c \) and \( g(x) = c x^{2} + b x + a \), with \( x \in [-1, 1] \). For any \( a, b, c \) such that \( |f(x)| \leq 1 \), find the maximum value of \( F(x) \) as \( a, b, \) and \( c \) vary.

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