
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
Let $\{a_{n}\}$ be a sequence of positive terms with the sum of the first $n$ terms denoted as $S_{n}$, and $a_{1}=1$. For $n\geqslant 2$, $a_{n}=\sqrt{{S_n}}+\sqrt{{S_{n-1}}}$. <br/>$(1)$ Find the general formula for the sequence $\{a_{n}\}$; <br/>$(2)$ Let $\{b_{n}\}$ be a sequence with $b_{1}=1$, and $b_{n+1}-b_{n}=2^{n-1}\cdot a_{n}$. Find the general formula for the sequence $\{b_{n}\}$.

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