
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 \( f(x)=x^{3}+b x^{2}+c x+d \) is an increasing function on \((-\infty, 0)\) and a decreasing function on \([0,2]\), and the equation \( f(x)=0 \) has three roots, namely \(\alpha, 2, \beta\), find the range of values for \( |\alpha-\beta| \).

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