
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 the sequence $\{a\_n\}$ satisfying $a\_1=1$, $\sqrt{a_{n+1}+3}=\sqrt{a\_n+3}+2$, $n\in\mathbb{N}^*$.

(1) Find the general term formula for the sequence $\{a\_n\}$.
(2) Let $\{b\_n\}$ be a geometric sequence with common ratio $2$ that satisfies $4\log \_{2}b_{n}\cdot\log \_{2}b_{n+1}=a\_{n}+12n+11(n\in\mathbb{N}^*)$, find the sum of the first $n$ terms, $S\_n$, for the sequence $\{b\_n-\log \_{2}b_{n}\}$.

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