
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 an arithmetic sequence $\{a_n\}$ with a common difference not equal to $0$, and a geometric sequence $\{b_n\}$ satisfying: $a_1=b_1=1$, $a_2=b_2$, $2a_3-b_3=1$.  
$(1)$ Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$.  
$(2)$ Let the sum of the first $n$ terms of the sequence $\{\log_{3}^{b_n}\}$ be $S_n$, find $S_n$.

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