
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 a common difference greater than $0$ and satisfies $a_3a_6=55,a_2+a_7=16$.

(1) Find the general term formula for the sequence $\{a_n\}$.

(2) The geometric sequence $\{b_n\}$ satisfies $b_1=a_1,b_2=a_2-1$. If we define a new sequence $\{c_n=a_n \cdot b_n\}$, find the sum of the first $n$ terms of the sequence $\{c_n\}$, denoted as $S_n$.

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