
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 arithmetic sequence $\{a_{n}\}$ and geometric sequence $\{b_{n}\}$ that satisfy: $a_{1}=b_{1}=1$, $a_{2}=b_{2}$, $2a_{3}-b_{3}=1$.

(I) Find the general term formulas for the sequences $\{a_{n}\}$ and $\{b_{n}\}$;

(II) Let $c_{n}=a_{n}b_{n}$, find the sum $S_{n}$ of the first $n$ terms of the sequence $\{c_{n}\}$.

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