
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 an arithmetic sequence with the sum of its first $n$ terms being $S\_n$. The sequence $\{b\_n\}$ is a geometric sequence with a common ratio greater than 0. It is known that $b\_1 = -2a\_1 = 2$, and $a\_3 + b\_2 = -1$, $S\_3 + 2b\_3 = 7$.

1. Find the general term formulas for both sequences $\{a\_n\}$ and $\{b\_n\}$.
2. Define a new sequence $\{C\_n\}$ as follows: $C\_n = \begin{cases} 2, & \text{if $n$ is odd} \\ \frac{-2a\_n}{b\_n}, & \text{if $n$ is even} \end{cases}$. Calculate the sum of the first $2n$ terms of this sequence, denoted as $T_{2n}$.

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