
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
Find all real numbers \(x\) for which there exists a triangle \(ABC\) with circumradius 2, such that \(\angle ABC \geq 90^{\circ}\), and
\[ x^{4} + a x^{3} + b x^{2} + c x + 1 = 0 \]
where \(a = BC\), \(b = CA\), and \(c = AB\).

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