
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 the sum of the first $n$ terms as $S_{n}$, $a_{1}=2$, and $S_{5}=20$.  
$(1)$ Find the general formula for the sequence $\{a_{n}\}$.  
$(2)$ If a geometric sequence $\{b_{n}\}$ has a common ratio of $q=\frac{1}{2}$ and satisfies $a_{4}+b_{4}=9$, find the sum of the first $n$ terms of the sequence $\{a_{n}-b_{n}\}$.

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