
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 $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, and a geometric sequence $\{b_n\}$ satisfying $a_1=3$, $b_1=1$, $b_2+S_2=10$, and $a_5-2b_2=a_3$:

$(1)$ Find the general formulas for sequences $\{a_n\}$ and $\{b_n\}$.

$(2)$ Let $c_n=\begin{cases} \frac{2}{S_n}, & n=2m-1 \ (m\in \mathbb{N}^*) \\ b_n, & n=2m \ (m\in \mathbb{N}^*) \end{cases}$, and let $T_n$ be the sum of the first $n$ terms of sequence $\{c_n\}$. Find $T_n$.

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