
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 the sequence $\{a_n\}$ is an arithmetic sequence, and its sum of the first $n$ terms is $S_n$, $\{b_n\}$ is a geometric sequence, and $a_1=b_1=2$, $a_4+b_4=27$, $S_4-b_4=10$.  
(1) Find the general formulas for sequences $\{a_n\}$ and $\{b_n\}$.  
(2) Let $T_n=a_1b_1+a_2b_2+\ldots+a_nb_n$, where $n\in\mathbb{N}^*$, find $T_n$ (for $n\in\mathbb{N}^*$, $n\geq2$).

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