
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 $n$, if the arithmetic sequence $a\_1$, $a\_2$, $a\_3$, $…$ satisfies $a\_1^{2}+a\_{2n+1}^{2}\leqslant 10$, then the maximum value of $S=a\_{2n+1}+a\_{2n+2}+a\_{2n+3}+…+a\_{4n+1}$ is $\_\_\_\_\_\_$.

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