
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 $\{a_n\}$ is an arithmetic sequence with the sum of the first $n$ terms denoted as $S_n$, and $\{b_n\}$ is a geometric sequence, where $a_1=b_1=2$ and $a_4+b_4=27$.  
(1) Determine the general formulas for the sequences $\{a_n\}$ and $\{b_n\}$.  
(2) Define $T_n=a_nb_1+a_{n-1}b_2+\ldots+a_1b_n$, where $n\in\mathbb{N}^*$, and calculate $T_n$.

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