
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 function $f(x)=\sqrt{3}\sin{\omega}x\cos{\omega}x-\cos^2{\omega}x+\frac{1}{2}$ with a minimum positive period of $6$.<br/>$(1)$ Given that in triangle $ABC$, the opposite sides of angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively, with $c > b$. If $f\left(\frac{6}{\pi}A\right)=1$ and $f\left(\frac{6}{\pi}B\right)=\frac{1}{2}$, find the value of $\frac{c}{b}$;<br/>$(2)$ If ${a_n}=n\cdot f\left(2n+\frac{1}{2}\right)$, find the sum of the first $2022$ terms of the sequence $\{a_{n}\}$ denoted by $S_{2022}$.

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