
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 an arithmetic sequence $\{a_n\}$ with the sum of the first $n$ terms denoted by $S_n$, where $S_3=6$ and $S_6=21$:  
(1) Find the general formula for the $n^{th}$ term of the sequence $\{a_n\}$.  
(2) Calculate the sum of the first $n$ terms of the sequence $\{2^n a_n\}$, denoted by $T_n$.

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