
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 sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, $a_1=1$, $a_{n+1}=2S_n+1$ ($n\in\mathbb{N}^*$), and an arithmetic sequence $\{b_n\}$ where $b_n>0$ ($n\in\mathbb{N}^*$), and $b_1+b_2+b_3=15$. Also, $a_1+b_1$, $a_2+b_2$, $a_3+b_3$ form a geometric sequence.
(Ⅰ) Find the general formula for sequences $\{a_n\}$ and $\{b_n\}$;
(Ⅱ) 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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