
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
The sum of the first $n$ terms of the arithmetic sequence $\{a_n\}$ is $S_n$, and the sequence $\{b_n\}$ is a geometric sequence, satisfying $a_1=3$, $b_1=1$, $b_2+S_2=10$, and $a_5-2b_2=a_3$.

(1) Find the general formulas for the sequences $\{a_n\}$ and $\{b_n\}$.

(2) Let $c_n=\begin{cases} \frac{2}{S_n}, & \text{if } n \text{ is odd}, \\ b_n, & \text{if } n \text{ is even}, \end{cases}$ and let the sum of the first $n$ terms of the sequence $\{c_n\}$ be $T_n$. Find $T_{2n}$.

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