
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 two sequences $\{a_n\}$ and $\{b_n\}$ satisfying $a_1=b_1=1$, $a_2=3$, and $S_{n+1}+S_{n-1}=2(S_n+1)$ for $n\geqslant 2$, $n\in\mathbb{N}^*$, where $S_n$ is the sum of the first $n$ terms of the sequence $\{a_n\}$. Also, for any $n\in\mathbb{N}^*$, it holds that $b_1+2b_2+2^2b_3+\ldots+2^{n-2}b_{n-1}+2^{n-1}b_n=a_n$.

(1) Find the general formula for the 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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