
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 sequence $\{a_{n}\}$ is a geometric sequence with a common ratio of $q$, the sum of the first $n$ terms is $S_{n}$, and it satisfies $a_{1}+a_{3}=2q+1$, $S_{3}=3a_{2}+1$.  
$(1)$ Find the general formula for the sequence $\{a_{n}\}$;  
$(2)$ If the sequence $\{b_{n}\}$ satisfies $b_{n}=\left\{\begin{array}{l}{{a}_{n+1}-{a}_{n}, \text{for odd } n}\\{\frac{3{a}_{n}}{4{a}_{n}^{2}-5{a}_{n}+1}, \text{for even } n}\end{array}\right.$, 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}
user