
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 it satisfies $S_n + \frac{1}{2}a_n = 1$ for $n \in \mathbb{N}^*$,

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

$(2)$ Let $b_n = \log_{3}(1-S_{n+1})$ for $n \in \mathbb{N}^*$, find the value of the positive integer $n$ that satisfies the equation $\frac{1}{b_1b_2} + \frac{1}{b_2b_3} + \cdots + \frac{1}{b_nb_{n+1}} = \frac{25}{51}$.

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