
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 $S_{n}$, and ${a}_{1}=\frac{1}{2}$, $2n(S_{n+1}-S_{n})=(n+1){a}_{n}$.  
$(1)$ Find the general formula for $\{a_{n}\}$;  
$(2)$ Let ${b}_{n}=n(2-S_{n})$, where $n \in \mathbb{N}^{*}$. If $b_{n} \leqslant \lambda$ holds for any $n \in \mathbb{N}^{*}$, find the range of real number $\lambda$.

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