
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
Consider the set of all equations of the form $x^3 + b_2x^2 + b_1x + b_0 = 0$, where $b_2$ and $b_0$ are real constants with $|b_i| \leq 3$ for $i = 0, 2$ and $b_1$ is a real constant with $|b_1| \leq 5$. Let $s$ be the largest positive real number which satisfies at least one of these equations. Determine the possible range for the largest positive real number s.

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