
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
In an increasing geometric sequence $\{a_{n}\}$, where $a_{1}\cdot a_{2}=8$ and $a_{1}+a_{2}=6$, and $n\in N^{*}$. <br/>$(1)$ Find the general term formula of the sequence $\{a_{n}\}$; <br/>$(2)$ If $b_{n}=2a_{n}+3$, find the sum of the first $n$ terms of the sequence $\{b_{n}\}$ denoted as $T_{n}$.

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