
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 point $M$ is a moving point on the circle with center $E$ and equation $\left( x+\sqrt{3} \right)^{2}+y^{2}=16$, and point $F\left( \sqrt{3},0 \right)$, the perpendicular bisector of segment $MF$ intersects $EM$ at point $P$.

$(1)$ Find the equation of the trajectory $C$ of point $P$;

$(2)$ The sides of rectangle $ABCD$ are tangent to curve $C$, and the area of rectangle $ABCD$ is denoted as $S$. Find the range of $S$.

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