
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 an ellipse $C:\frac{{{x^2}}}{4}+{y^2}=1$ with left and right foci $F_{1}$ and $F_{2}$, point $E$ is a moving point on the ellipse $C$, and $O$ is the origin.  
$(1)$ If $∠{F_1}E{F_2}=\frac{π}{3}$, find the area of $\triangle F_{1}EF_{2}$.  
$(2)$ If the line $l$ with slope $k$ passing through point $E$ intersects the ellipse $C$ at another point $F$, $OE\bot OF$, $M$ is the midpoint of segment $EF$, and ray $OM$ intersects the ellipse $C$ at point $N$, the areas of $\triangle NMF$ and $\triangle EMO$ are denoted as $S_{1}$ and $S_{2}$, find the range of $\frac{{{S_1}}}{{{S_2}}}$.

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