
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 $\{a_{n}\}$ where $a_{1}=2$, ${a}_{1}+\frac{{a}_{2}}{2}+\frac{{a}_{3}}{3}+\cdots+\frac{{a}_{n}}{n}={a}_{n+1}-2$, $n\in N^{*}$, and a sequence $\{b_{n}\}$ where $b_{1}=2$, $b_{2}=4$, ${b}_{n+1}^{2}={b}_{n}{b}_{n+2}$, $n\in N^{*}$.  
$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$;  
$(2)$ Let a sequence $\{c_{n}\}$ satisfy ${c}_{n}=\frac{{a}_{n}}{{b}_{n}}$, the sum of the first $n$ terms of sequence $\{c_{n}\}$ is $S_{n}$, and the inequality ${(-1)}^{n}\lambda < {S}_{n}+\frac{n}{{2}^{n-1}}$ holds for all $n\in N^{*}$, find the range of real number $\lambda$.

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