
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 an arithmetic sequence $\{a\_n\}$ with the sum of the first $n$ terms $S\_n$, $a\_2=\frac{1}{8}$, and the sequence $S\_1+\frac{1}{16}$, $S\_2$, $S\_3$ forms an arithmetic sequence. Additionally, sequence $\{b\_n\}$ satisfies $b\_n=2n$.

1. Find the general term formula for sequence $\{a\_n\}$.
2. If $c\_n=a\_n b\_n$, find the sum of the first $n$ terms $T\_n$ for sequence $\{c\_n\}$.

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