
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 \(2n\) real numbers \(a_{1}, a_{2}, \cdots, a_{2n}\) satisfying the condition
$$
\sum_{i=1}^{2 n-1}\left(a_{i+1}-a_{i}\right)^{2}=1,
$$
find the maximum value of \(\left(a_{n+1}+a_{n+2}+\cdots+a_{2n}\right) - \left(a_{1}+a_{2}+\cdots+a_{n}\right)\).

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