
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 $q \gt 1$, if $a_{2}+a_{3}+a_{4}=14$, and $a_{2}$, $a_{3}+1$, $a_{4}$ are the $1$st, $3$rd, and $5$th terms of an arithmetic sequence $\{b_{n}\}$.<br/>$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$;<br/>$(2)$ If ${c}_{n}=\frac{2{a}_{n}}{({a}_{n+1}-1)({a}_{n+2}-1)}+\frac{{b}_{n}}{{a}_{n}}$, find the sum of the first $n$ terms of sequence $\{c_{n}\}$ denoted as $S_{n}$.

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