
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 an arithmetic sequence $\{a_{n}\}$ with the sum of the first $n$ terms as $S_{n}(n\in N_{+})$, and a geometric sequence $\{b_{n}\}$, where $a_{1}=3$, $b_{1}=1$, $b_{2}+S_{2}=10$, and $a_{5}-2b_{2}=a_{3}$.<br/>$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$;<br/>$(2)$ Let ${c_n}=\left\{\begin{array}{l}\frac{2}{{{S_n}}},n\text{ is odd}\\ 2{a_n}{b_n},n\text{ is even}\end{array}\right.$, and the sum of the first $n$ terms of sequence $\{c_{n}\}$ be $T_{n}$, find $T_{2n}$.

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