
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 a sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, with $a_1 = 4$, and for every positive integer $n$, the following equation holds: $$\frac{1}{2} S_n - a_n + 2 = 0.$$
(1) Find the general term formula for the sequence $\{a_n\}$.
(2) Let $b_n = a_n \cdot \log_{\frac{1}{2}} \left(\frac{1}{a_n}\right)$. Find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$.

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