
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 the arithmetic sequence $\{a_{n}\}$, it is known that $a_{1}+a_{5}=14$ and $a_{6}=16$.<br/>$(1)$ Find the general formula for $\{a_{n}\}$;<br/>$(2)$ If $\{b_{n}\}$ is a sequence of positive geometric progression, and $b_{6}=b_{1}b_{5}=64$, 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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