
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 \( M \) be a positive integer. It is known that whenever \( \left|a x^{2}+b x+c\right| \leq 1 \) for all \( |x| \leq 1 \), then \( |2 a x+b| \leq M \) for all \( |x| \leq 1 \). Find the smallest possible value of \( M \).

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