
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 $a_n=2-2S_n$ for $n\in\mathbb{N}^{*}$,

(1) Find the explicit formula for the $n$-th term $a_n$ of the sequence.

(2) Let another sequence $\{b_n\}$ be defined such that $b_n=\log_3(1-S_n)$ for $n\in\mathbb{N}^{*}$. If $\dfrac{1}{{b_2}{b_3}}+\dfrac{1}{{b_3}{b_4}}+\ldots+\dfrac{1}{{b_n}{b_{n+1}}} =\dfrac{25}{51}$, find the value of the natural number $n$.

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