
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 ratio of the distance from moving point $M$ to fixed point $F(-\sqrt{3},0)$ to its distance from the line $x=-\dfrac{4\sqrt{3}}{3}$ is

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

(2) Does there exist a point $E$ on $x$ such that for any line passing through $E$, if it intersects with curve $C$ at two points $M$ and $N$, then $\dfrac{1}{{\left|EM\right|}^{2}}+ \dfrac{1}{{\left|EN\right|}^{2}}$ is a constant? If so, find the coordinates of point $E$ and the corresponding constant.

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