
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 $\{ a_{n} \}$ that satisfies $a_{1}=1$, and $2a_{2}$, $a_{6}$, $a_{8}+1$ form a geometric sequence with a common ratio greater than $1$.
(1) Find the general term formula for the sequence $\{ a_{n} \}$.
(2) Let $b_{n}=\frac{a_{n}}{2^{n}}$, find the sum of the first $n$ terms, $T_{n}$, of the sequence $\{ b_{n} \}$.

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