
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
In the sequence $\{a_n\}$, we are given $a_1 = 1$, let $S_n$ be the sum of the first $n$ terms of the sequence $\{a_n\}$, and it satisfies $\frac{2a_n}{a_nS_n - S_n^2} = 1$ for $n \geq 2$.  
(1) Determine whether the sequence $\left\{ \frac{1}{S_n} \right\}$ is an arithmetic sequence, and provide the reason;  
(2) Find the general term formula for the sequence $\{a_n\}$;  
(3) Let $b_n = \begin{cases}
1, & n=1 \\
-\frac{2}{na_n}, & n\geq 2
\end{cases}$, and let $T_n = \frac{1}{b_1+n} + \frac{1}{b_2+n} + \ldots + \frac{1}{b_n+n}$. If $T_n < m$ holds for all $n\geq 2$, find the range of the real number $m$.

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