
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

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

holds true.

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

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