
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 general term of the sequence $\{a_{n}\}$ is $a_{n}=2^{n-1}$, and the sequence $\{b_{n}\}$ is an arithmetic sequence. Let $A=\{a_{1}$, $a_{2}$, $\ldots $, $a_{n}$, $\ldots \}$, $B=\{b_{1}$, $b_{2}$, $\ldots $, $b_{n}$, $\ldots \}(n\in N^{*})$, and the sequence formed by arranging the elements in the set $A\cup B$ in ascending order is denoted as $\{c_{n}\}$. <br/>$(1)$ If $c_{n}=n(n\in N^{*})$, write down a valid general term formula for $\{b_{n}\}$ and explain the reason; <br/>$(2)$ If $b_{n}=\sqrt{2}n,{d_n}={3^n}+{(-1)^n}{({\frac{{{b_n}}}{n}})^{2n}}⋅λ,({n∈{N^*}})$, and the sequence $\{d_{n}\}$ is strictly monotonically increasing for $n\in N^{*}$, find the range of real number $\lambda$; <br/>$(3)$ If $A\cap B=\varnothing $, the first $5$ terms of the sequence $\{c_{n}\}$ form a geometric sequence, and $c_{1}=1$,$c_{9}=8$, find all valid sequences $\{b_{n}\}$ that satisfy the conditions.

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