
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 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 terms of the sequences $\{a\_n\}$ and $\{b\_n\}$;

(2) Let $c=\begin{cases} \frac{2}{S\_n}, &(n\text{ is odd}) \\ b\_n, &(n\text{ is even})\end{cases}$, and denote the sum of the first $n$ terms of the sequence $\{c\_n\}$ as $T\_n$. Find $T\_{2n}$.

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