
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 $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\left(a > b > 0\right)$ with eccentricity $\frac{\sqrt{2}}{2}$, where three points $M_1(-2,\sqrt{2})$, $M_2(2,-\sqrt{2})$, $M_3(\sqrt{2},\frac{\sqrt{3}}{2})$ have exactly two points lying on the ellipse.<br/>$(Ⅰ)$ Find the equation of ellipse $C$;<br/>$(Ⅱ)$ If the top vertex of $C$ is $E$ and the right focus is $F$, a line passing through point $F$ intersects $C$ at points $A$ and $B$ (excluding the vertex of the ellipse). Lines $EA$ and $EB$ intersect the line $x-y-4=0$ at points $P$ and $Q$, find the minimum area of $\triangle EPQ$.

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