
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 the function \( f(x) = x^2 + ax + b \), where \( a \) and \( b \) are real numbers, let \( M(a, b) \) be the maximum value of \( |f(x)| \) on the interval \([-1,1]\). When \( M(a, b) \leq 2 \), find the maximum value of \( |a| + |b| \).

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