
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 \(a, b, c\) be distinct nonzero real numbers such that the equations
\[
\begin{array}{l}
ax^3 + bx + c = 0, \\
bx^3 + cx + a = 0, \\
cx^3 + ax + b = 0
\end{array}
\]
have a common root, and among these three equations, there exist two equations without imaginary roots. Find the minimum value of \(\frac{a}{b} + \frac{b}{c} + \frac{c}{a}\).

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