
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 a geometric sequence $\{a_n\}$ with a common ratio $q=2$ and the sum of the first $3$ terms is $7$, and an arithmetic sequence $\{b_n\}$ satisfies $b_1=3$, $2b_2=a_2+a_4$.  
$(1)$ Find the general formula for sequences $\{a_n\}$ and $\{b_n\}$.  
$(2)$ Find the sum of the first $n$ terms $S_n$ of the sequence $\left\{ \frac{2}{(2n-1)b_n} \right\}$.

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