
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
In the arithmetic sequence $\{a\_n\}$, $a\_1=10$, the common difference $d$ is an integer, and $a\_4a\_5<0$. In the sequence $\{b\_n\}$, $$b_{1}=2，b_{n+1}=b_{n}+2^{n}$$.
(1) Find the general term formulas for the sequences $\{a\_n\}$ and $\{b\_n\}$;
(2) If $c\_n=a\_n b\_n$, find the sum of the first $n$ terms $S\_n$ of the sequence $\{c\_n\}$.

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