
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
The real sequence \(\left\{a_{n}\right\}\) satisfies the conditions: \(a_{1}=\frac{\sqrt{2}}{2}+1\), \(a_{2}=\sqrt{2}+1\), and \(a_{n+1}+a_{n-1}=\frac{n}{a_{n}-a_{n-1}}+2\) for \(n \geq 2\). Find the general term \(a_{n}\) for \(n \geq 1\).

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