
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 $\{a_n\}$, $a_1=1$, and for $n\geq2$, the sum of its first $n$ terms $S_n$ satisfies $S_n^2=a_n\left(S_n-\frac{1}{2}\right)$  
(1) Find the expression for $S_n$;  
(2) Let $b_n=\frac{S_n}{2n+1}$, and the sum of the first $n$ terms of the sequence $\{b_n\}$ be $T_n$, find $\lim_{n\to\infty}T_n$.

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