
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 sequence $\left\{a_{n}\right\}$ with terms that are all greater than or equal to 1, satisfying:
$$
a_{1}=1, \quad a_{2}=1+\frac{\sqrt{2}}{2}, \quad\left(\frac{a_{n}}{a_{n+1}-1}\right)^{2}+\left(\frac{a_{n}-1}{a_{n-1}}\right)^{2}=2 .
$$

Find:
(1) The general term formula of the sequence $\left\{a_{n}\right\}$;
(2) The value of $\lim _{n \rightarrow+\infty} \frac{a_{n}}{n}$.

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