
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
In the complex plane, $O$ is the origin, and the complex number $z_{1}=m+i$ is a root of the equation ${x^2}-2\sqrt{3}x+n=0$ with respect to $x$ $\left(m,n\in R\right)$.  
$(1)$ Find the values of $m$ and $n$;  
$(2)$ If the complex number $z_{2}=1+\sqrt{3}i$, and the points corresponding to $z_{1}$, $z_{2}$, and $\frac{z_{2}}{z_{1}}$ are $A$, $B$, and $C$ respectively, let $\triangle AOB$ have an area of $S_{1}$, and $\triangle BOC$ have an area of $S_{2}$, find $\frac{S_{1}}{S_{2}}$.

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