
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
In the geometric sequence $\{a\_n\}$, $a\_1=2$, and $a\_2+1$ is the arithmetic mean of $a\_1$ and $a\_3$.
(I) Find the general term formula of $\{a\_n\}$ and the sum of the first $n$ terms $S\_n$;
(II) Given that $\{b\_n\}$ is an arithmetic sequence, $T\_n$ is the sum of its first $n$ terms, $b\_2=a\_1$, and $b\_8=a\_2+a\_4$, find $T\_n$.

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