
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 geometric sequence $\{a_{n}\}$, given that $a_{1}=1$ and $a_{2}$ is the arithmetic mean of $a_{1}$ and $a_{3}-1$.  
$(1)$ Find the general formula for the sequence $\{a_{n}\}$;  
$(2)$ If the sequence $\{b_{n}\}$ satisfies $b_{n}=2n+a_{n}$ $(n\in N^{*})$, find the sum of the first $n$ terms of the sequence $\{b_{n}\}$ denoted as $S_{n}$.

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