
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 that the sum of the first $n$ terms of the sequence $\{a_{n}\}$ is $S_{n}$, with $a_{1}=0$ and $a_{n} \gt 0$ for $n\geqslant 2$, and $a_{n+1}=\sqrt{S_{n+1}}+\sqrt{S_{n}$. Find:<br/>
$(1)$ The general formula for the sequence $\{a_{n}\}$;<br/>
$(2)$ If the sequence $\{b_{n}\}$ satisfies $b_{n}=\frac{a_{n}}{2^{n}}$, and the sum of the first $n$ terms of $\{b_{n}\}$, denoted as $T_{n}$, is always less than $m$, find the smallest integer value of $m$.

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