
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 $a\_nb\__{n+1}-b\__{n+1}=nb\_n$ for all $n∈N^{*}$,
(I) find the general term formula for $\{b\_n\}$;
(II) let $c\_n=a\_nb\_n$, find the sum of the first $n$ terms $S\_n$ of the sequence $\{c\_n\}$.

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