
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 $\{a_n\}$ with the sum of the first $n$ terms as $S_n$ and $S_n+ \frac{1}{2}a_n=1$ for $n \in \mathbf{N}^+$.  
(1) Find the general term formula for the sequence $\{a_n\}$.  
(2) Let $b_n=\log_{\frac{1}{3}}(1-S_n)$ for $n \in \mathbf{N}^+$, find the value of $\frac{1}{b_1b_2} + \frac{1}{b_2b_3} + \ldots + \frac{1}{b_nb_{n+1}}$.

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