
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 ellipse $C_{1}: \frac{{x}^{2}}{8} + \frac{{y}^{2}}{4} = 1$ and another ellipse $C_{2}$ with the same eccentricity as $C_{1}$. The foci of ellipse $C_{2}$ lie on the $y$-axis and pass through the point $(1, \sqrt{2})$.  
$(1)$ Find the standard equation of ellipse $C_{2}$.  
$(2)$ Let $A$ be the upper vertex of ellipse $C_{1}$. The line $l$ passing through the origin intersects ellipse $C_{2}$ at points $P$ and $Q$. Let $M$ and $N$ be the other intersection points of lines $AP$ and $AQ$ with ellipse $C_{1}$, respectively. If the areas of triangles $\triangle AMN$ and $\triangle APQ$ are denoted by $S_{1}$ and $ $S_{2}$, respectively, find the range of $\frac{{S}_{1}}{{S}_{2}}$.

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