
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 a positive geometric sequence with the first term $a_{1}=1$, and the sequence $\{b_{n}\}$ is an arithmetic sequence with a common difference $d=2$, satisfying $b_{3}=2a_{2}$ and $a_{3}=b_{4}+1$. Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$. If $c_{n}=$____, find the sum of the first $n$ terms $S_{n}$ of the sequence $\{c_{n}\}$. Choose one of the following conditions to complete the blank line above and provide the solution.

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