
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 $S_{n}=2^{n}-1$, and the set $A=\{k|S_{m}=\log _{2}a_{k}$, $m\in N^{*}$, $k\in N^{*}\}$.  
$(1)$ Find the set $A$;  
$(2)$ If $b_{n}=\left\{\begin{array}{l}{a}_{n},n∈A,\\ lo{g}_{2}{a}_{n},n∉A,\end{array}\right.$, find the sum of the first $30$ terms of the sequence $\{b_{n}\}$.

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