
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 $S_n$. The sequence $S_1+a_1$, $S_2+a_2$, $S_3+a_3$ forms an arithmetic sequence.

(1) Find the general term formula for the sequence $\{a_n\}$.

(2) If $b_n=a_n\cdot \log_2 2a_n$, and the sum of the first $n$ terms of the sequence $\{b_n\}$ is denoted as $T_n$, 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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