
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
Given that \(a\), \(b\), and \(c\) are three distinct real numbers, and in the quadratic equations

\[
x^{2} + ax + b = 0
\]

\[
x^{2} + bx + c = 0
\]

\[
x^{2} + cx + a = 0
\]

any two of the equations have exactly one common root, find the value of \(a^{2} + b^{2} + c^{2}\).

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