
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 first term of a geometric sequence $\{a\_n\}$ is $\frac{1}{2}$, and the sequence is decreasing. The sum of the first $n$ terms is denoted as $S\_n$. Also, $S\_1+a\_1$, $S\_2+a\_2$, $S\_3+a\_3$ form an arithmetic sequence.

(I) Find the general term formula for the sequence $\{a\_n\}$;
(II) If $b\_n=a\_n\cdot \log\_2 a\_n$, and the sum of the first $n$ terms of the sequence $\{b\_n\}$ is $T\_n$, find the maximum value of $n$ that satisfies the inequality $\frac{T\_n+2}{n+2}\geqslant \frac{1}{16}$.

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