
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
Let \(a\), \(b\), and \(c\) be distinct non-zero real numbers such that the equations
\[a x^{3}+b x+c=0,\]
\[b x^{3}+c x+a=0,\]
\[c x^{3}+a x+b=0\]
have a common root, and among these three equations, there exist two equations with no complex roots. Find the minimum value of \(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\).

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