
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 that \( x_{1}, x_{2}, \cdots, x_{10} \) are all positive integers and \( x_{1} + x_{2} + \cdots + x_{10} = 2005 \), find the maximum and minimum values of \( x_{1}^{2} + x_{2}^{2} + \cdots + x_{10}^{2} \).

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