
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 a sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_{n}=-a_{n}-\left(\frac{1}{2}\right)^{n-1}+2$ (where $n$ is a positive integer).
(1) Find the general term formula for the sequence $\{a_n\}$.
(2) Let $c_{n}=\frac{n+1}{n}a_{n}$, and $T_{n}=c_{1}+c_{2}+\ldots+c_{n}$, find the value of $T_{n}$.

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