
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 a moving point $M$ to a fixed point $F(-\sqrt{3},0)$ and its distance to the line $x=-\dfrac{4\sqrt{3}}{3}$ is a constant $\dfrac{\sqrt{3}}{2}$, denote the trajectory of the moving point $M$ as curve $C$.

(1) Find the equation of curve $C$;

(2) If line $l_{1}$: $y=kx+t_{1}$ intersects curve $C$ at two distinct points $A$, $B$, line $l_{2}$: $y=kx+t_{2}(t_{1} \neq t_{2})$ intersects curve $C$ at two distinct points $D$, $E$, and $|AB|=|DE|$. Find the maximum value of the area $S$ of the convex quadrilateral with vertices $A$, $B$, $D$, $E$.

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