
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
All values of the quadratic polynomial \( f(x) = a x^2 + b x + c \) on the interval \([0, 2]\) do not exceed 1 in absolute value. What is the maximum possible value of \( |a| + |b| + |c| \) in this case? For which function \( f(x) \) is this value achieved?

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