
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 a geometric sequence with $a_1=1$ and $a_4=27$; $S_n$ is the sum of the first $n$ terms of the arithmetic sequence $\{b_n\}$, with $b_1=3$ and $S_5=35$.

$(1)$ Find the general formula for $\{a_n\}$ and $\{b_n\}$.

$(2)$ Let the sequence $\{c_n\}$ satisfy $c_n=a_nb_n$ ($n\in \mathbb{N}^*$), find the sum of the first $n$ terms of the sequence $\{c_n\}$, denoted as $T_n$.

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