
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
The sequence $\{a_{n}\}$ satisfies $a_{1}=1$, $a_{n}=\frac{n{a}_{n-1}}{{a}_{n-1}+2n-2}$ ($n\geqslant 2$, $n\in N^{*}$).<br/>$(1)$ Find the values of $a_{2}$, $a_{3}$, and $a_{4}$;<br/>$(2)$ Find the general formula for the sequence $\{a_{n}\}$;<br/>$(3)$ Let $b_{n}=(1-\frac{1}{{2}^{n}})a_{n}$. Find the sum of the first $n$ terms of the sequence $\{b_{n}\}$.

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