
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 $D(x_{0},y_{0})$ is a point on the circle $O: x^{2}+y^{2}=12$, and $E(x_{0},0)$, the moving point $P$ satisfies $\overrightarrow{OP}= \frac {1}{2} \overrightarrow{ED}+ \frac { \sqrt {3}}{3} \overrightarrow{OE}$. Let the trajectory of the moving point $P$ be curve $C$.
$(1)$ Find the equation of curve $C$;
$(2)$ If the moving line $l: y=kx+m$ is tangent to curve $C$, and perpendiculars are drawn from points $A_{1}(-2,0)$ and $A_{2}(2,0)$ to line $l$ at points $M$ and $N$ respectively, is there a maximum or minimum value for the area of quadrilateral $A_{1}MNA_{2}$? If yes, find the maximum or minimum value and the value of $k$ at that time; if no, explain why.

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