
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
Determine the largest positive real number r which satisfies at least one of the equations \(x^3 + b_2x^2 + b_1x + b_0 = 0\), where \(b_2\), \(b_1\), \(b_0\) are integers and \(|b_i| \leq 3\) for \(i = 0,1,2\).

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