
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 $S_{n}$ is the sum of the first $n$ terms of the sequence $\{a_n\}$, with $a_{1} < 2$, $a_n > 0$, and $6S_{n}= a_{n}^{2}+3a_{n}+2$ for $n \in \mathbb{N^*}$.  
$(1)$ Find the general formula for the sequence $\{a_n\}$.  
$(2)$ If for every $n \in \mathbb{N^*}$, $b_n=(-1)^{n} a_{n}^{2}$, 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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