
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 sequence $\{a_{n}\}$ is an arithmetic sequence with a non-zero common difference, satisfying $a_{1}=1$, $a_{4}+a_{5}=a_{9}$, the sum of the first $n$ terms of the positive sequence $\{b_{n}\}$ is $S_{n}$, and $S_n=3^n-1$.  
$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$;  
$(2)$ Insert $1$ number $c_{11}$ between $b_{1}$ and $b_{2}$ to form an arithmetic sequence $b_{1}$, $c_{11}$, $b_{2}$; insert $2$ numbers $c_{21}$, $c_{22}$ between $b_{2}$ and $b_{3}$ to form an arithmetic sequence $b_{2}$, $c_{21}$, $c_{22}$, $b_{3}$; $\ldots$; insert $n$ numbers $c_{n1}$, $c_{n2}$, $\cdots$, $c_{nn}$ between $b_{n}$ and $b_{n+1}$ to form an arithmetic sequence $b_{n}$, $c_{n1}$, $c_{n2}$, $\cdots$, $c_{nn}$, $b_{n+1}$.  
$($ⅰ) Find $c_{nk}$;  
$($ⅱ) Find the value of $c_{11}+c_{21}+c_{22}+\cdots +c_{n1}+c_{n2}+\cdots +c_{nn}$.

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