
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
Determine the largest constant \(C\) such that for all real numbers \(x_{1}, x_{2}, \ldots, x_{6}\), the inequality

\[
\left(x_{1}+x_{2}+\cdots+x_{6}\right)^{2} \geq C \cdot \left(x_{1}(x_{2}+x_{3})+x_{2}(x_{3}+x_{4})+\cdots+x_{6}(x_{1}+x_{2})\right)
\]

holds.

Find, for this \(C\), all \(x_{1}, x_{2}, \ldots, x_{6}\) for which equality holds.

(Walther Janous)

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