
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
Let \(a_i \in \mathbb{R}\) (for \(i = 1, 2, \cdots, n\)). If the sum of any two of these numbers is non-negative, then for any non-negative real numbers \(x_i\) (for \(i = 1, 2, \cdots, n\)) that satisfy \(\sum_{i=1}^{n} x_{i}=1\), the inequality \(\sum_{i=1}^{n} a_{i} x_{i} \geqslant \sum_{i=1}^{n} a_{i} x_{i}^{2}\) holds.

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