
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 $\{a_n\}$, the sum of the first $n$ terms is $S_n$. The sequence $\{b_n\}$ is a geometric sequence with a common ratio greater than $0$, and $b_1 = -2a_1 = 2$, $a_3 + b_2 = -1$, and $S_3 + 2b_3 = 7$.
1. Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$.
2. Let $c_n = \begin{cases} 2 & \text{if } n \text{ is odd,} \\ \frac{-2a_n}{b_n} & \text{if } n \text{ is even,} \end{cases}$ find the sum of the first $n$ terms of the sequence $\{c_n\}$, denoted as $T_n$.

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