
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]


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Given \( n \geq 3 \), \(\omega = \cos \frac{2\pi}{n} + \mathrm{i} \sin \frac{2\pi}{n}\) is an \( n \)-th root of unity. Also, \( x_i \) (for \( i = 0, 1, \cdots, n-1 \)) are real numbers with \( x_0 \geq x_1 \geq \cdots \geq x_{n-1} \). Find the necessary and sufficient conditions that \( x_0, x_1, \cdots, x_{n-1} \) must satisfy for the following equation to hold:
\[ x_0 + x_1 \omega + x_2 \omega^2 + \cdots + x_{n-1} \omega^{n-1} = 0. \]

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