
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, the sum of its first $n$ terms is $S_n$, and the sequence $\{b_n\}$ is a geometric sequence with a common ratio greater than $0$, and $b_1=-2a_1=2$, $a_3+b_2=-1$, $S_3+2b_3=7$.

$(1)$ 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 $\{c_n\}$, $T_n$.

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