
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 the sequence $\{a_n\}$, with $a_1 = \sqrt{2} + 1$ and $\frac{a_n - \sqrt{n}}{a_{n-1} - \sqrt{n-1}} = \sqrt{1 + \frac{1}{n}}$ for $n = 2, 3, \ldots$, let $b_n = \frac{1}{a_n}$. Denote the sum of the first $n$ terms of the sequence $\{b_n\}$ as $S_n$.
(1) Find the general term formula of the sequence $\{a_n\}$;
(2) If for any $n \in \mathbb{N}^*$, $\sqrt{n+2} \leq \lambda (S_n + 1) \leq n + 101$ always holds, find the range of the real number $\lambda$.

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