
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 a geometric series with positive terms {a<sub>n</sub>} (n ∈ N*), where the first term a<sub>1</sub> is 3 and the sum of the first n terms is S<sub>n</sub>. It is known that S<sub>3</sub>+a<sub>3</sub>, S<sub>5</sub>+a<sub>5</sub>, and S<sub>4</sub>+a<sub>4</sub> form an arithmetic sequence.
(1) Find the general term formula for the sequence {a<sub>n</sub>}.
(2) Find the sum of the first n terms T<sub>n</sub> for the sequence {nS<sub>n</sub>}.

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