
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
Let the sum of the first $n$ terms of the sequence $\{a_{n}\}$ be $S_{n}$, and it is known that ${S}_{n}=\frac{1}{2}(3{a}_{n}-1)$ for $n\in\mathbb{N}^{*}$.  
$(1)$ Find the general formula for $\{a_{n}\}$;  
$(2)$ Let ${b}_{n}=\left\{\begin{array}{l}n+{a}_{n}, \text{ for odd } n\\ n⋅{a}_{n}, \text{ for even } n\end{array}\right.$. Find the sum of the first $2n$ terms of the sequence $\{b_{n}\}$, denoted as $T_{2n}$.

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