
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 an geometric sequence $\{a_n\}$ with the first term $\frac{1}{2}$, which is a decreasing sequence. The sum of the first $n$ terms is $S_n$, and $S_1+a_1$, $S_2+a_2$, $S_3+a_3$ form an arithmetic sequence.

(Ⅰ) Find the general 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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