
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 \( n \geq 2 \) be a positive integer, and let the real numbers \( a_1, a_2, \ldots, a_n \) satisfy the equation \( \sum_{i=1}^{n} \left| a_i \right| + \left| \sum_{i=1}^{n} a_i \right| = 1 \). Find the minimum and maximum values of \( \sum_{i=1}^{n} a_i^2 \).

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