
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
Given that the real numbers \(a\), \(b\), and \(c\) satisfy \(\left|a x^{2} + b x + c\right|\) with a maximum value of 1 on the interval \(x \in [-1,1]\), what is the maximum possible value of \(\left|c x^{2} + b x + a\right|\) on the interval \(x \in [-1,1]\)?

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