
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 positive integer \(n\) and a positive number \(M\), for all arithmetic sequences \(a_1, a_2, a_3, \cdots\) that satisfy the condition \(a_1^2 + a_{n+1}^2 \leq M\), find the maximum value of \(S = a_{n+1} + a_{n+2} + \cdots + a_{2n+1}\).

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