
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
Consider the arithmetic sequence $\{a\_n\}$ with the sum of its first $n$ terms denoted as $S\_n$. 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 $C\_n = \begin{cases} \frac{2}{S\_n}, & n \text{ is odd} \\ b\_n, & n \text{ is even} \end{cases}$. Define the sequence $\{c\_n\}$ with the sum of its first $n$ terms denoted as $T\_n$. Find $T\_{2n}$.

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