
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, $\{b\_n\}$ is a geometric sequence, and $a\_1=b\_1=2$, $b\_4=54$, $a\_1+a\_2+a\_3=b\_2+b\_3$.

1. Find the general term formula for both sequences $\{a\_n\}$ and $\{b\_n\}$.
2. The sequence $\{c\_n\}$ satisfies $c\_n=a\_n b\_n$. Find the sum of the first $n$ terms, denoted as $S\_n$, for the sequence $\{c\_n\}$.

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