
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 number sequence \( \{a_n\} \) 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 \( n \geq 2 \). Determine the general term formula \( a_n = \frac{n}{\sqrt{2}} + 1 \) for \( n \geq 1 \).

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