
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 a geometric sequence $\{a_{n}\}$, all terms are positive, and $a_{1}+a_{3}=10$, $4{a}_{3}^{2}=a_{2}\cdot a_{6}$.<br/>$(1)$ Find the general term formula for the sequence $\{a_{n}\}$;<br/>$(2)$ Find the sum of the first $n$ terms of the sequence $\{n\cdot a_{n}\}$ denoted by $T_{n}$.

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