
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 that the sequence $\{a_n\}$ is an arithmetic sequence with a common difference of $2$, and the sequence $\{b_n\}$ satisfies $(b_1=1, b_2=\frac{1}{2})$, if $a_n b_{n+1} - b_{n+1} = n b_n$ holds for all $n\in\mathbb{N}^*$:

(1) Find the general formula for the $n$-th term of $\{b_n\}$;

(2) Let $\{c_n=\frac{1}{a_n a_{n+1}}\}$, find the sum of the first $n$ terms of $\{c_n\}$, denoted as $S_n$.

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