
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 \(a_{1}, a_{2}, \cdots, a_{2009}\) be non-negative real numbers satisfying the following conditions:
(1) \(a_{1}+a_{2}+\cdots+a_{2009}=2\);
(2) \(a_{1} a_{2}+a_{2} a_{3}+\cdots+a_{2008} a_{2009}+a_{2009} a_{1}=1\).

Find the maximum and minimum values of \(S=a_{1}^{2}+a_{2}^{2}+\cdots+a_{2009}^{2}\).

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