
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
There are \(2n+1\) players \(A_1, A_2, \cdots, A_{2n+1}\) in a chess round-robin tournament. Each player competes with the other \(2n\) players once, and there are no draws. Let \(w_i\) be the number of games won by player \(A_i\). Find the minimum and maximum values of \(S=\sum_{i=1}^{2n+1} w_i^2\).

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