
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, $S\_n$ is the sum of the first $n$ terms of $\{a\_n\}$, and $a\_10=19$, $S\_10=100$. For any positive integer $n$, the sequence $\{b\_n\}$ always satisfies $b\_1·b\_2·b\_3…b\_n−1=a\_n+2$.

(1) Find the general term formulas for the sequences $\{a\_n\}$ and $\{b\_n\}$.

(2) Let $c\_n=(-1)^n \frac{4n\cdot b\_n}{(2n+1)^2}$, find the sum $T\_n$ of the first $n$ terms of the sequence $\{c\_n\}$.

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