
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 $n \geqslant 2$, 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 $b_n = \frac{2^n}{S_n}$, find the sum of the first $n$ terms of $\{b_n\}$, denoted as $T_n$.

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