
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 common difference of $d\left(d \gt 0\right)$, $a_{1}=2$, and $a_{3}+2$, $a_{4}$, $a_{6}-4$ form a geometric sequence. Also, the sequence $\{b_{n}\}$ satisfies $b_{1}+b_{2}+b_{3}+\cdots +b_{n}=d\cdot b_{n}-d$, $n\in N^{*}$.  
$(1)$ Find the general formula for the sequence $\{b_{n}\}$;  
$(2)$ Let $b_{0}=1$, when $n\in [b_{k-1}$, $b_{k})$, ${c}_{n}=k(k∈{N}^{*})$, find the sum of the first $2^{n}-1$ terms of the sequence $\{c_{n}\}$.

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