
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 $C:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with left and right foci $F_{1}$ and $F_{2}$, and eccentricity $\frac{\sqrt{2}}{2}$. Let $P$ be a point on the ellipse $C$, and the maximum area of $\triangle PF_{1}F_{2}$ is $4$.  
$(Ⅰ)$ Find the equation of the ellipse $C$;  
$(Ⅱ)$ Through point $F_{2}$, draw two perpendicular lines $AB$ and $DE$, where $A$, $B$, $D$, $E$ are all on the ellipse $C$. Find the range of $\frac{|DE|}{|AB|}$.

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