
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 sequence $\{a_{n}\}$ where $a_{1}=1$ and $a_{2}=3$, and a geometric sequence $\{b_{n}\}$ with a common ratio $q \gt 0$, satisfying $2b_{n}(a_{n+1}-a_{n})=b_{n+2}$. Find:<br/>
$(1)$ The general formula for the sequence $\{a_{n}\}$;<br/>
$(2)$ Let the sum of the first $n$ terms of the sequence $\{b_{n}\}$ be $S_{n}$. If ______, let the sequence $\{c_{n}\}$ be defined as ${c_n}=\left\{{\begin{array}{l}{{a_n},n \text{ is odd}}\\{{b_n},n \text{ is even}}\end{array}}\right.$. Find the sum of the first $2n$ terms of the sequence $\{c_{n}\}$. Choose one of the conditions from the three given conditions: $S_{2}+1=\frac{1}{2}S_{3}$, $b_{1}$, $2a_{2}-1$, $b_{3}$ form an arithmetic sequence, $S_{7}=254$, and solve it.<br/>
Note: If multiple conditions are chosen and solved separately, the score will be based on the first solution.

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