
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 \in \mathbb{Z}^\plus{}$  and let  $ a, b \in \mathbb{R}.$  Determine the range of  $ x_0$  for which

\[ \sum^n_{i\equal{}0} x_i \equal{} a \text{ and } \sum^n_{i\equal{}0} x^2_i \equal{} b,\]

where  $ x_0, x_1, \ldots , x_n$  are real variables.

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