
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 $S_n$, and $S_n + \frac{1}{2}a_n = 1$ ($n \in \mathbb{N}^+$),
(Ⅰ) Find the general formula for the sequence $\{a_n\}$;
(Ⅱ) Let $b_n = \log_{\frac{1}{3}}(1 - S_{n+1})$ ($n \in \mathbb{N}^+$), and define $T_n = \frac{1}{b_1 b_2} + \frac{1}{b_2 b_3} + \dots + \frac{1}{b_n b_{n+1}}$, find $T_n$.

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