
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 that the sequence $\{a_{n}\}$ is an increasing arithmetic sequence, and the sequence $\{b_{n}\}$ is a geometric sequence, with $a_{1}=3$, $a_{1}-1$, $a_{2}-1$, $a_{3}+1$ forming a geometric sequence, and $b_{1}=1$, $a_{5}-2b_{2}=a_{3}$. <br/>$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$; <br/>$(2)$ If ${c}_{n}={b}_{n}+\log_{2}\frac{{a}_{n}}{{a}_{n+1}}$, find the sum of the first $n$ terms of the sequence $\{c_{n}\}$ denoted as $S_{n}$.

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