
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.

Also, find for this \( C \) all \( x_1, x_2, \ldots, x_6 \) for which equality holds.

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