
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 $\{a_{n}\}$ with the sum of the first $n$ terms denoted as $S_{n}$, where ${a_1}=\frac{1}{2}$, and for $n\geqslant 2$, ${S}_{n}^{2}={a}_{n}{S}_{n}-{a}_{n}$. Find:<br/>
$(1)$ $S_{n}$;<br/>
$(2)$ Let the sum of the first $n$ terms of the sequence $\left\{{\frac{{{2^n}}}{{{S_n}}}}\right\}$ be $T_{n}$. If $λ{T_n}≤({{n^2}+9})•{2^n}$ always holds, find the range of values for $\lambda$.

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