
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 the following conditions:①$a_{3}=5$,$a_{2}+a_{5}=6b_{2}$；②$b_{2}=2$,$a_{3}+a_{4}=3b_{3}$；③$S_{3}=9$,$a_{4}+a_{5}=8b_{2}$, choose one of these three conditions to complete the following question and solve it.<br/>For an arithmetic sequence $\{a_{n}\}$ with a common difference of $d\left(d \gt 1\right)$, and a sum of the first $n$ terms $S_{n}$, and a geometric sequence $\{b_{n}\}$ with a common ratio of $q$, where $a_{1}=b_{1}$, $d=q$,____.<br/>$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$.<br/>$(2)$ Let ${c_n}=\frac{{{a_n}}}{{{b_n}}}$, find the sum of the first $n$ terms of the sequence $\{c_{n}\}$.

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