
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]


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Given real numbers \( a, b, c \), and the functions \( f(x)=a x^{2}+b x+c \) and \( g(x)=a x+b \), find \( f(x) \) if the maximum value of \( g(x) \) is 2 when \(-1 \leq x \leq 1\).

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