
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 that $\{a_n\}$ is an arithmetic sequence, the sum of its first $n$ terms is $S_n$, $\{b_n\}$ is a geometric sequence $(b_n > 0)$, and $a_1=b_1=2$, $a_3+b_3=16$, $S_4+b_3=34$.  
(1) Find the general formula for sequences $\{a_n\}$ and $\{b_n\}$.  
(2) Let $T_n$ be the sum of the first $n$ terms of the sequence $\{a_nb_n\}$, find $T_n$.

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