
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 $\{a_{n}\}$ is an increasing arithmetic sequence, and $a_{4}$ and $a_{6}$ are two roots of the equation $x^{2}-7x+12=0$; the sum of the first $n$ terms of the sequence $\{b_{n}\}$ is $S_{n}$, and $S_{n}=2b_{n}-2$ for $n\in N^{*}$.  
$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$;  
$(2)$ Find the sum of the first $n$ terms of the sequence $\{a_{n}\cdot b_{n}\}$, denoted as $T_{n}$.

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