
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 $\{a\_n\}$ where each term is positive, let $S\_n$ denote the sum of its first $n$ terms. For any $n∈N^{∗}$, it is known that $2S_{n}=a_{n}^{2}+a_{n}$.
(1) Find the general formula for the sequence $\{a\_n\}$.
(2) Another sequence $\{b\_n\}$ is defined by $b_{1}=1$ and $2b_{n+1}-b_{n}=0$ for $n∈N^{∗}$. Let $c_{n}=a_{n}b_{n}$. Find the sum of the first $n$ terms, denoted as $T_{n}$, for the sequence $\{c\_n\}$.

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