
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 $\{a_{n}\}$ with the sum of the first $n$ terms as $S_{n}$, $a_{1}=3$, $a_{n+1}=2S_{n}+3$; in the sequence $\{b_{n}\}$, $b_{1}=1$, $b_{n}-b_{n+1}+1=0$.  
$(1)$ Find the general formulas $a_{n}$ and $b_{n}$ for sequences $\{a_{n}\}$ and $\{b_{n}\}$, respectively.  
$(2)$ Let $c_{n}=a_{n}b_{n}$. 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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