
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 $\{a\_n\}$ satisfies $a\_1=2$, $a_{n+1}-2a_{n}=2$, and the sequence $b_{n}=\log _{2}(a_{n}+2)$. If $S_{n}$ is the sum of the first $n$ terms of the sequence $\{b_{n}\}$, then the minimum value of $\{\frac{S_{n}+4}{n}\}$ is ___.

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