
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 $\{a_n\}$, the sum of the first $S_n$ terms is $S_n$, with $a_1=3$. Sequence $\{b_n\}$ is a geometric sequence, and $b_1=1$, $b_n>0$, $b_2+S_2=10$, $S_5=5b_3+3a_2$, $n\in\mathbb{N}^*$.  
(1) Find the general formula for sequences $\{a_n\}$ and $\{b_n\}$.  
(2) Find the sum of the first $n$ terms $T_n$ of the sequence $\{a_n\cdot b_n\}$.

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