
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 formula of the sequence $\{a\_n\}$ is $a\_n=n$, the general term formula of $\{b\_n\}$ is $b\_n=2^{n}$, and the value of $c\_n$ is the number of elements in the first $n$ terms of $\{a\_n\}$ that contain elements of $\{b\_n\}$. Let $S\_n$ denote the sum of the first $n$ terms of the sequence $\{c\_n\}$. The correct statement(s) among the following is/are \_\_\_\_\_\_ (fill in the blank with the number(s) of all correct conclusions).

$①$ When $n=2^{k}(k=1,2,3…)$, $c\_n=k$;

$②$ When $n=2^{k+1}-1(k=1,2,3…)$, $c\_n=k$;

$③$ When $n=2^{k+1}-1(k=1,2,3…)$, $S\_n=(k-1)⋅2^{k+1}+2$.

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