
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 $\{a_{n}\}$ where $a_{n} \gt 0$, and ${a}_{n+1}^{2}=a_{n}a_{n+1}+2{a}_{n}^{2}$, and $3a_{1}$, $a_{2}+3$, $a_{3}$ form an arithmetic sequence.<br/>$(1)$ Find the general formula for $\{a_{n}\}$;<br/>$(2)$ Let $b_{n}=\left\{\begin{array}{l}{a_n},n{ is odd}\\{log_{\frac{1}{2}}}{a_n},n{ is even}\end{array}\right.$, 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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