
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 \( a_i \in \mathbf{R} \) (for \( i = 1, 2, \cdots, 10 \)) and \( \sum_{i=1}^{10} \frac{a_i^2}{a_i^2 + 1} = 1 \), find the range of values for \( \sum_{i=1}^{10} \frac{a_i}{a_i^2 + 1} \).

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