
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 \( n \) be a positive integer and let \( a, b \) be given real numbers. Determine the range of \( x_{0} \) for which
$$
\sum_{i=0}^{n} x_{i} = a \text { and } \sum_{i=0}^{n} x_{i}^{2} = b,
$$
where \( x_{0}, x_{1}, \ldots, x_{n} \) are real variables.

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