
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
Let the positive geometric sequence $\{a_n\}$ have the first term $a_1= \frac{1}{2}$, and let the sum of the first $n$ terms be $S_n$. Given that $2^{10}S_{30}-(2^{10}+1)S_{20}+S_{10}=0$, find:  
(1) The general term of $\{a_n\}$;  
(2) The sum of the first $n$ terms of $\{nS_n\}$, denoted as $T_n$.

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