
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
In the arithmetic sequence $\{a_n\}(n\in\mathbb{N}^*)$, it is known that $a_1=2$, $a_5=6$.  
$(1)$ Find the common difference $d$ and the general term $a_n$  
$(2)$ Let $b_n=2^{a_n}(n\in\mathbb{N}^*)$, calculate the sum of the first $n$ terms of the sequence $\{b_n\}$, $S_n$

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