
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 that the sum of the first $n$ terms of a geometric sequence $\{a_{n}\}$ is $S_{n}$, $a_{1}=1$, and $4S_{n}$, $3S_{n+1}$, $2S_{n+2}$ form an arithmetic sequence.  
$(1)$ Find the general formula for $\{a_{n}\}$;  
$(2)$ If the sequence $\{b_{n}\}$ satisfies $b_{n}=\left\{\begin{array}{l}{a}_{n},\text{ for odd } n\\ n-1,\text{ for even } n\end{array}\right.$, find the sum of the first $2n$ terms of $\{b_{n}\}$, denoted as $T_{2n}$.

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