
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 $E:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ passing through points $M(-\sqrt{2},\frac{2\sqrt{6}}{3})$ and $N(\sqrt{3},\sqrt{2})$.  
$(1)$ Find the equation of $E$;  
$(2)$ A line passing through a point $P$ on the $x$-axis, not parallel to the coordinate axes, intersects $E$ at points $A$ and $B$, where the perpendicular bisector of segment $AB$ intersects the $x$-axis at point $C$, and $|AB|=2\sqrt{3}|PC|$. Find the coordinates of point $P$.

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