
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 that the trajectory of point $M$ in the plane is an Apollonian circle (the ratio of the distances between the moving point $M$ and the two fixed points $A$ and $B$ is $\frac{|MA|}{|MB|}=λ(\lambda\ \ \gt 0$, $\lambda \neq 1$, and $\lambda$ is a constant), with the equation $x^{2}+y^{2}=4$, where the fixed points are the right focus $F$ and the right vertex $A$ of an ellipse $C:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\left(a \gt b \gt 0\right)$, and the major axis of the ellipse $C$ is $4\sqrt{2}$.<br/>$(1)$ Find the standard equation of the ellipse $C$;<br/>$(2)$ Let the left focus of the ellipse $C$ be $E$, and a line $l$ passing through point $A$ intersects the circle $x^{2}+y^{2}=4$ at points $S$ and $T$. Find the maximum area of $\triangle EST$.

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