
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 $S_{n}$, $a_{1}=1$, $\frac{S_{n}}{n}-\frac{S_{n-1}}{n-1}=2$ for $n\geq 2$. Find the general formula for the sequence $\{a_{n}\}$; Let $b_{n}=\frac{a_{n}}{2^n}$, find the sum of the first $n$ terms of the sequence $\{b_{n}\}$ denoted as $T_{n}$.

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