
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 and $\{b\_n\}$ is a geometric sequence, with $a\_1=1$, $b\_1=2$, $a\_2+b\_2=7$, and $a\_3+b\_3=13$.

1. Find the general term formula for both $\{a\_n\}$ and $\{b\_n\}$.
2. If $$c_{n}= \begin{cases} \overset{a_{n},n\text{ is odd}}{b_{n},n\text{ is even}}\end{cases}$$, find the sum of the first $2n$ terms of the sequence $\{c\_n\}$, denoted as $S\_2n$.

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