
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 $\{a_{n}\}$ that satisfies $a_{1}=\frac{1}{2}$ and $a_{n+1}=2a_{n}+\frac{1}{2}$, and a sequence $\{b_{n}\}$ with the sum of its first $n$ terms as $S_{n}=n^{2}+n$. <br/>$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$; <br/>$(2)$ Let $\{c_{n}\}$ be a sequence such that $c_{n}=a_{n}\cdot b_{n}$. Find the sum of the first $n$ terms of the sequence $\{c_{n}\}$ denoted as $T_{n}$.

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