
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 \(x^3 + a_2x^2 + a_1x + a_0 = 0\), where \(a_2\), \(a_1\), \(a_0\) are real constants with \(|a_i| \leq 3\) for \(i = 0,1,2\) and the sum \(a_2 + a_1 + a_0 = -6\). Find the largest positive real number \(r\) which satisfies at least one of these equations.
A) \( 1 < r < 2 \)
B) \( 2 < r < 3 \)
C) \( 3 < r < 4 \)
D) \( 4 < r < 5 \)

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