
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 sum of the first $n$ terms of the sequence $\{a\_n\}$ is $S\_n= \frac {1}{2}n(n-1)$, and $a\_n$ is the middle term of an arithmetic sequence with first term $1$ and last term $b_n$.
(1) Find the general term formula for the sequences $\{a\_n\}$ and $\{b\_n\}$;
(2) If $c\_n= \frac {1}{a\_n(n+1)}(n\geqslant 2)$, find the sum of $c\_2+c\_3+c\_4+…+c\_n$.

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