
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 $\{a_n\}$ with the sum of the first $n$ terms denoted as $S_n$, $a_1 > 0$, $a_1 \cdot a_2= \frac{3}{2}$, $S_5=10$.  
$(1)$ Find the general formula for the sequence $\{a_n\}$.  
$(2)$ Let the sequence $b_n= \begin{cases} 2^{a_n}, & \text{if } n \text{ is odd} \\ a_n, & \text{if } n \text{ is even} \end{cases}$, find the sum of the first $2n+1$ terms of $\{b_n\}$, denoted as $T_{2n+1}$.

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