
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 a positive integer $n \geqslant 2$, find the maximum value of the constant $C(n)$ such that for all real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying $x_{i} \in (0, 1)$ for $i = 1, 2, \cdots, n$ and
$$
\left(1 - x_{i}\right)\left(1 - x_{j}\right) \geqslant \frac{1}{4} \quad \text{for} \quad 1 \leqslant i < j \leqslant n,
$$
the following inequality holds
$$
\sum_{i=1}^{n} x_{i} \geqslant C(n) \sum_{1 \leqslant i < j \leqslant n}\left(2x_{i}x_{j} + \sqrt{x_{i}x_{j}}\right).
$$

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