
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 $x^3 + b_2x^2 + b_1x + b_0 + c_1x^4 = 0$, where $b_2$, $b_1$, $b_0$ are real constants and $|b_i| < 3$ for $i = 0, 1, 2$, and $c_1 = 1$. Let $r$ be the largest positive real number which satisfies at least one of these equations. Determine the range of $r$.

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