
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 a sequence $\{a_n\}$ that satisfies: $a_{n+1}-a_{n}=d$ ($n\in\mathbb{N}^*$), and the sum of the first $n$ terms is denoted as $S_n$, with $a_{1}=4$, $S_{3}=21$.  
$(1)$ Find the general formula for the sequence $\{a_n\}$.  
$(2)$ Suppose another sequence $\{b_n\}$ satisfies $b_1= \frac{16}{7}$, $b_{n+1}-b_n=2^{a_n}$, find the general formula for the sequence $\{b_n\}$.

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