
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 $a_1=2$, $b_1=1$, $a_{n+1}=2a_n (n\in\mathbb{N}^*)$, and $b_1+ \frac{1}{2}b_2+ \frac{1}{3}b_3+\ldots+ \frac{1}{n}b_n=b_{n+1}-1 (n\in\mathbb{N}^*)$  
(I) Find $a_n$ and $b_n$;  
(II) Let the sum of the first $n$ terms of the sequence $\{a_nb_n\}$ be $T_n$, find $T_n$.

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